real analysis
Contents
3.3. real analysis#
Some notes on real analysis, following the textbook Understanding analysis
3.3.1. ch 1 - the real numbers#
there is no rational number whose square is 2 (proof by contradiction)
contrapositive: $\(-q \to -p\)$ - logically equivalent
triangle inequality: \(\|a+b\| \leq \|a\| + \|b\|\) (often use |a-b| = |(a-c)+(c-b)|)
axiom of completeness - every nonempty set \(A \subseteq \mathbb{R}\) that is bounded above has a least upper bound
doesn’t work for \(\mathbb{Q}\)
supremum = supA = least upper bound (similarly, infimum)
supA is an upper bound of A
if \(s \in \mathbb{R}\) is another u.b. then \(s \geq supA\)
can be restated as \(\forall \epsilon > 0, \exists a \in A\) \(s-\epsilon < a\)
nested interval property - for each \(n \in N\), assume we are given a closed interval \(I_n = [a_n,b_n]=\{ x \in \mathbb{R} : a_n \leq x \leq b_n \}\) Assume also that each \(I_n\) contains \(I_{n+1}\). Then, the resulting nested sequence of nonempty closed intervals \(I_1 \supseteq I_2 \supseteq ...\) has a nonempty intersection use AoC with x = sup{\(a_n: n \in \mathbb{N}\)} in the intersection of all sets
archimedean property
\(\mathbb{N}\) is unbounded above (sup \(\mathbb{N}=\infty\))
\(\forall x \in \mathbb{R}, x>0, \exists n \in \mathbb{N}, 0 < \frac{1}{n} < x\)
\(\mathbb{Q}\) is dense in \(\mathbb{R}\) - for every \(a,b \in \mathbb{R}, a<b\), \(\exists r \in \mathbb{Q}\) s.t. \(a<r<b\)
pf: want \(a < \frac{m}{n} < b\)
by Archimedean property, want \(\frac{1}{n} < b-a\)
corollary: the irrationals are dense in \(\mathbb{R}\)
there exists a real number \(r \in \mathbb{R}\) satisfying \(r^2 = 2\)
pf: let r = \(sup \{ t \in \mathbb{R} : t^2 < 2 \}\). disprove \(r^2<2, r^2>2\) by considering \(r+\frac{1}{n},r-\frac{1}{n}\)
A ~ B if there exists f:A->B that is 1-1 and onto
A is finite - there exists n \(\in \mathbb{N}\) s.t. \(\mathbb{N}_n\)~A
countable = \(\mathbb{N}\)~A.
uncountable - inifinite set that isn’t countable
Q is countable
pf: Let \(A_n = \{ \pm \frac{p}{q}:\) where p,q \(\in \mathbb{N}\) are in lowest terms with p+q=n}
R is uncountable
pf: Assume we can enumerate \(\mathbb{R}\) Use NIP to exclude one point from \(\mathbb{R}\) each time. The intersection is still nonempty, so we didn’t succesfully enumerate \(\mathbb{R}\)
\(\frac{x}{x^2-1}\) maps (0,1) \(\to \mathbb{R}\)
countable union of countable sets is countable
if \(A \subseteq B\) and B countable, then A is either countable or finite
if \(A_n\) is a countable set for each \(n \in \mathbb{N}\), then their union is countable
the open interval (0,1) = \(\{ x \in \mathbb{R} : 0 < x < 1 \}\) is uncountable
pf: diagonalization - assume there exists a function from (0,1) to \(\mathbb{R}\). List the decimal expansions of these as rows of a matrix. Complement of diagonal does not exist.
cantor’s thm - Given any set A, there does not exist a function f:\(A \to P(A)\) that is onto
P(A) is the set of all subsets of A
3.3.2. ch 2 - sequences and series#
a sequence \((a_n)\) converges to a real number if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall n\geq N, \|a_n-a\| < \epsilon\)
otherwise it diverges
if a limit exists, it is unique
a sequence \((x_n)\) is bounded if there exists a number M > 0 such that \(\|x_n\|\leq M \forall n \in \mathbb{N}\)
every convergent sequence is bounded
algebraic limit thm - let lim \(a_n = a\) and lim \(b_n\) = b. Then
lim(\(ca_n\)) = ca
lim(\(a_n+b_n\)) = a+b
lim(\(a_n b_n\)) = ab
lim(\(a_n/b_n\)) = a/b, provided b \(\neq\) 0
pf 3: use triangle inequality, \(\|a_nb_n-ab\|=\|a_nb_n-ab_n+ab_n-ab\|=...=\|b_n\|\|a_n-a\|+\|a\|\|b_n-b\|\)
pf 4: show \((b_n) \to b\) implies \((\frac{1}{b_n}) \to \frac{1}{b}\)
order limit thm - Assume lim \(a_n = a\) and lim \(b_n\) = b.
If \(a_n \geq 0\) \(\forall n \in \mathbb{N}\), then \(a \geq 0\)
If \(a_n \leq b_n\) \(\forall n \in \mathbb{N}\), then \(a \leq b\)
If \(\exists c \in \mathbb{R}\) for which \(c \leq b_n\) \(\forall n \in \mathbb{N}\), then \(c \leq b\)
pf 1: by contradiction
monotone - increasing or decreasing (not strictly)
monotone convergence thm - if a sequence is monotone and bounded, then it converges
convergence of a series
define \(s_m=a_1+a_2+...+a_m\)
\(\sum_{n=1}^\infty a_n\) converges to A \(\iff (s_m)\) converges to A
cauchy condensation test - suppose \(a_n\) is decreasing and satisfies \(a_n \geq 0\) for all \(n \in \mathbb{N}\). Then, the series \(\sum_{n=1}^\infty a_n\) converges iff the series \(\sum_{n=1}^\infty 2^na_{2^n}\) converges
p-series \(\sum_{n=1}^\infty 1/n^p\) converges iff p > 1
3.3.2.1. 2.5#
let \((a_n)\) be a sequence and \(n_1<n_2<...\) be an increasing sequence of natural numbers. Then \((a_{n_1},a_{n_2},...)\) is a subsequence of \((a_n)\)
subsequences of a convergent sequence converge to the same limit as the original sequence
can be used as divergence criterion
bolzano-weierstrass thm - every bounded sequence contains a convergent subsequence
pf: use NIP, keep splitting interval into two
3.3.2.2. 2.6#
\((a_n)\) is a cauchy sequence if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall m,n\geq N, \|a_n-a_m\| < \epsilon\)
cauchy criterion - a sequence converges \(\iff\) it is a cauchy sequence
cauchy sequences are bounded
overview: AoC \(\iff\) NIP \(\iff\) MCT \(\iff\) BW \(\iff\) CC
3.3.2.3. 2.7#
algebraic limit thm - let \(\sum_{n=1}^\infty a_n\) = A, \(\sum_{n=1}^\infty b_n\) = B
\(\sum_{n=1}^\infty ca_n\) = cA
\(\sum_{n=1}^\infty a_n+b_n\) = A+B
cauchy criterion for series - series converges \(\iff\) \((s_m)\) is a cauchy sequence
if the series \(\sum_{n=1}^\infty a_n\) converges then lim \(a_n=0\)
comparison test
geometric series - \(\sum_{n=0}^\infty a r^n = \frac{a}{1-r}\)
\(s_m = a+ar+...+ar^{m-1} = \frac{a(1-r^m)}{1-r}\)
absolute convergence test
alternating series test
decreasing
lim \(a_n\) = 0
then, \(\sum_{n=1}^\infty (-1)^{n+1} a_n\) converges
rearrangements: there exists one-to-one correspondence
if a series converges absolutely, any rearrangement converges to same limit
3.3.3. ch 3 - basic topology of R#
3.3.3.1. 3.1 cantor set#
C has small length, but its cardinality is uncountable
discussion of dimensions, doubling sizes leads to 2^dimension sizes
Cantor set is about dimension .631
3.3.3.2. 3.2 open/closed sets#
A set O \(\subseteq \mathbb{R}\) is open if for all points a \(\in\) O there exists an \(\epsilon\)-neighborhood \(V_{\epsilon}(a) \subseteq O\)
\(V_{\epsilon}(a)=\{ x \in R : \|x-a\| < \epsilon\)}
the union of an arbitrary collection of open sets is open
the intersection of a finite collection of open sets is open
a point x is a limit point of a set A if every \(\epsilon\)-neighborhood \(V_{\epsilon}(x)\) of x intersects the set A at some point other than x
a point x is a limit point of a set A if and only if x = lim \(a_n\) for some sequence (\(a_n\)) contained in A satisfying \(a_n \neq x\) for all n \(\in\) N
isolated point - not a limit point
set \(F \subseteq \mathbb{R}\) closed - contains all limit points
closed iff every Cauchy sequence contained in F has a limit that is also an element of F
density of Q in R - for every \(y \in \mathbb{R}\), there exists a sequence of rational numbers that converges to y
closure - set with its limit points
closure \(\bar{A}\) is smallest closed set containing A
iff set open, complement is closed
R and \(\emptyset\) are both open and closed
the union of a finite collection of closed sets is closed
the intersection of an arbitrary collection of closed sets is closed
3.3.3.3. 3.3#
a set K \(\subseteq \mathbb{R}\) is compact if every sequence in K has a subsequence that converges to a limit that is also in K
Nested Compact Set Property - intersection of nested sequence of nonempty compact sets is not empty
let A \(\subseteq \mathbb{R}\). open cover for A is a (possibly infinite) collection of open sets whose union contains the set A.
given an open cover for A, a finite subcover is a finite sub-collection of open sets from the original open cover whose union still manages to completely contain A
Heine-Borel thm - let K \(\subseteq \mathbb{R}\). All of the following are equivalent
K is compact
K is closed and bounded
every open cover for K has a finite subcover
3.3.4. ch 4 - functional limits and continuity#
3.3.4.1. 4.1#
dirichlet function: 1 if r \(\in \mathbb{Q}\) 0 otherwise
3.3.4.2. 4.2 functional limits#
def 1. Let f:\(A \to R\), and let c be a limit point of the domain A. We say that \(lim_{x \to c} f(x) = L\) provided that for all \(\epsilon\) > 0, there exists a \(\delta\) > 0 s.t. whenever 0 < |x-c| < \(\delta\) (and x \(\in\) A) it follows that |f(x)-L|< \(\epsilon\)
def 2. Let f:\(A \to R\), and let c be a limit point of the domain A. We say that \(lim_{x \to c} f(x) = L\) provided that for every \(\epsilon\)-neighborhood \(V_{\epsilon}(L)\) of L, there exists a \(\delta\)-neighborhood \(V_{\delta}(\)c) around c with the property that for all x \(\in V_{\delta}(\)c) different from c (with x \(\in\) A) it follows that f(x) \(\in V_{\epsilon}(L)\).
sequential criterion for functional limits - Given function f:\(A \to R\) and a limit point c of A, the following 2 statements are equivalent:
\(lim_{x \to c} f(x) = L\)
for all sequences \((x_n) \subseteq\) A satisfying \(x_n \neq\) c and \((x_n) \to c\), it follows that \(f(x_n) \to L\).
algebraic limit thm for functional limits
divergence criterion for functional limits
3.3.4.3. 4.3 continuous functions#
a function f:\(A \to R\) is continuous at a point c \(\in\) A if, for all \(\epsilon\)>0, there exists a \(\delta\)>0 such that whenever |x-c|<\(\delta\) (and x\(\in\) A) it follows that \(\|f(x)-f( c)\|<\epsilon\). F is continous if it is continuous at every point in the domain A
characterizations of continuouty
criterion for discontinuity
algebraic continuity theorem
if f is continuous at c and g is continous at f( c) then g \(\circ\) f is continuous at c
3.3.4.4. 4.4 continuous functions on compact sets#
preservation of compact sets - if f continuous and K compact, then f(K) is compact as well
extreme value theorem - if f if continuous on a compact set K, then f attains a maximum and minimum value. In other words, there exist \(x_0,x_1 \in K\) such that \(f(x_0) \leq f(x) \leq f(x_1)\) for all x \(\in\) K
f is uniformly continuous on A if for every \(\epsilon\)>0, there exists a \(\delta\)>0 such that for all x,y \(\in\) A, \(\|x-y\| < \delta \implies \|f(x)-f(y)\| < \epsilon\)
a function f fails to be uniformly continuous on A iff there exists a particular \(\epsilon_o\) > 0 and two sequences \((x_n),(y_n)\) in A sastisfying \(\|x_n - y_n\| \to 0\) but \(\|f(x_n)-f(y_n)\| \geq \epsilon_o\)
a function that is continuous on a compact set K is uniformly continuous on K
3.3.4.5. 4.5 intermediate value theorem#
intermediate value theorem - Let f:[a,b]\( \to R\) be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c \(\in (a,b)\) where f( c) = L
a function f has the intermediate value property on an inverval [a,b] if for all x < y in [a,n] and all L between f(x) and f(y), it is always possible to find a point c \(\in (x,y)\) where f( c)=L.
3.3.5. ch 5 - the derivative#
3.3.5.1. 5.2 derivatives and the intermediate value property#
let g: A -> R be a function defined on an interval A. Given c \(\in\) A, the derivative of g at c is defined by g’( c) = \(\lim_{x \to c} \frac{g(x) - g( c)}{x-c}\), provided this limit exists. Then g is differentiable at c. If g’ exists for all points in A, we say g is differentiable on A
identity: \(x^n-c^n = (x-c)(x^{n-1}+cx^{n-2}+c^2x^{n-3}+...+c^{n-1}\))
differentiable \(\implies\) continuous
algebraic differentiability theorem
adding
scalar multiplying
product rule
quotient rule
chain rule: let f:A-> R and g:B->R satisfy f(A)\(\subseteq\) B so that the composition g \(\circ\) f is defined. If f is differentiable at c in A and g differentiable at f( c) in B, then g \(\circ\) f is differnetiable at c with (g\(\circ\)f)’( c)=g’(f( c))*f’( c)
interior extremum thm - let f be differentiable on an open interval (a,b). If f attains a maximum or minimum value at some point c \(\in\) (a,b), then f’( c) = 0.
Darboux’s thm - if f is differentiable on an interval [a,b], and a satisfies f’(a) < \(\alpha\) < f’(b) (or f’(a) > \(\alpha\) > f’(b)), then there exists a point c \(\in (a,b)\) where f’( c) = \(\alpha\)
derivative satisfies intermediate value property
3.3.5.2. 5.3 mean value theorems#
mean value theorem - if f:[a,b] -> R is continuous on [a,b] and differentiable on (a,b), then there exists a point c \(\in\) (a,b) where \(f'( c) = \frac{f(b)-f(a)}{b-a}\)
Rolle’s thm - f(a)=f(b) -> f’( c)=0
if f’(x) = 0 for all x in A, then f(x) = k for some constant k
if f and g are differentiable functions on an interval A and satisfy f’(x) = g’(x) for all x \(\in\) A, then f(x) = g(x) + k for some constant k
generalized mean value theorem - if f and g are continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c \(\in (a,b)\) where |f(b)-f(a)|g’( c) = |g(b)-g(a)|f’( c). If g’ is never 0 on (a,b), then can be restated \(\frac{f'( c)}{g'( c)} = \frac{f(b)-f(a)}{g(b)-g(a)}\)
given g: A -> R and a limit point c of A, we say that \(lim_{x \to c} g(x) = \infty\) if, for every M > 0, there exists a \(\delta\)> 0 such that whenever 0 < |x-c| < \(\delta\) it follows that g(x) ≥ M
L’Hospital’s Rule: 0/0 - let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of the point a. If f(a) = g(a) = 0 and g’(x) ≠0 for all x ≠a, then \(lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)
L’Hospital’s Rule: \(\infty / \infty\) - assume f and g are differentiable on (a,b) and g’(x) ≠0 for all x in (a,b). If \(lim_{x \to a} g(x) = \infty \), then \(lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)
3.3.6. ch 6 - sequences and series of function#
3.3.6.1. 6.2 uniform convergence of a sequence of functions#
for each n \(\in \mathbb{N}\) let \(f_n\) be a function defined on a set A\(\subseteq R\). The sequence (\(f_n\)) of functions converges pointwise on A to a function f if, for all x in A, the sequence of real numbers \(f_n(x)\) converges to f(x)
let (\(f_n\)) be a sequence of functions defined on a set A\(\subseteq\)R. Then (\(f_n\)) converges unformly on A to a limit function f defined on A if, for every \(\epsilon\)>0, there exists an N in \(\mathbb{N}\) such that \(\forall n ≥N, x \in A , \|f_n(x)-f(x)\|<\epsilon\)
Cauchy Criterion for uniform convergence - a sequence of functions \((f_n)\) defined on a set A \(\subseteq\) R converges uniformly on A iff \(\forall \epsilon > 0 \exists N \in \mathbb{N}\) s.t. whenever m,n ≥N and x in A, \(\|f_n(x)-f_m(x)\|<\epsilon\)
continuous limit thm - Let (\(f_n\)) be a sequence of functions defined on A that converges uniformly on A to a function f. If each \(f_n\) is continuous at c in A, then f is continuous at c
3.3.6.2. 6.3 uniform convergence and differentiation#
differentiable limit theorem - let \(f_n \to f\) pointwise on the closed interval [a,b], and assume that each \(f_n\) is differentiable. If \((f'_n)\) converges uniformly on [a,b] to a function g, then the function f is differentiable and f’=g
let (\(f_n\)) be a sequence of differentiable functions defined on the closed interval [a,b], and assume \((f'_n)\) converges uniformly to a function g on [a,b]. If there exists a point \(x_0 \in [a,b]\) for which \(f_n(x_0)\) is convergent, then (\(f_n\)) converges uniformly. Moreover, the limit function f = lim \(f_n\) is differentiable and satisfies f’ = g
3.3.6.3. 6.4 series of functions#
term-by-term continuity thm - let \(f_n\) be continuous functions defined on a set A \(\subseteq\) R and assume \(\sum f_n\) converges uniformly on A to a function f. Then, f is continuous on A.
term-by-term differentiability thm - let \(f_n\) be differentiable functions defined on an interval A, and assume \(\sum f'_n(x)\) converges uniformly to a limit g(x) on A. If there exists a point \(x_0 \in [a,b]\) where \(\sum f_n(x_0)\) converges, then the series \(\sum f_n(x)\) converges uniformly to a differentiable function f(x) satisfying f’(x) = g(x) on A. In other words, \(f(x) = \sum f_n(x)\) and \(f'(x) = \sum f'_n(x)\)
Cauchy Criterion for uniform convergence of series - A series \(\sum f_n\) converges uniformly on A iff \(\forall \epsilon > 0 \exists N \in N\) s.t. whenever n>m≥N, x in A \(\|f_{m+1}(x) + f_{m+2}(x) + f_{m+3}(x) + ...+f_n(x)\| < \epsilon\)
Wierstrass M-Test - For each n in N, let \(f_n\) be a function defined on a set A \(\subseteq\) R, and let \(M_n > 0\) be a real number satisfying \(\|f_n(x)\| ≤ M_n\) for all x in A. If \(\sum M_n\) converges, then \(\sum f_n\) converges uniformly on A
3.3.6.4. 6.5 power series#
power series f(x) = \(\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x_1 + a_2 x^2 + a_3 x^3 + ...\)
if a power series converges at some point \(x_0 \in \mathbb{R}\), then it converges absolutely for any x satisfying |x|<|\(x_0\)|
if a power series converges pointwise on the set A, then it converges uniformly on any compact set K \(\subseteq\) A
if a power series converges absolutely at a point \(x_0\), then it converges uniformly on the closed interval [-c,c], where c = |\(x_0\)|
Abel’s thm - if a power series converges at the point x = R > 0, the the series converges uniformly on the interval [0,R]. A similar result holds if the series converges at x = -R
if \(\sum_{n=0}^\infty a_n x^n\) converges for all x in (-R,R), then the differentiated series \(\sum_{n=0}^\infty n a_n x^{n-1}\) converges at each x in (-R,R) as well. Consequently the convergence is uniform on compact sets contained in (-R,R).
can take infinite derivatives
3.3.6.5. 6.6 taylor series#
Taylor’s Formula \(\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x_1 + a_2 x^2 + a_3 x^3 + ...\)
centered at 0: \(a_n = \frac{f^{(n)}(0)}{n!}\)
Lagrange’s Remainder thm - Let f be differentiable N+1 times on (-R,R), define \(a_n = \frac{f^{(n)}(0)}{n!}.....\)
not every infinitely differentiable function can be represented by its Taylor series (radius of convergence zero)
3.3.7. ch 7 - the Riemann Integral#
3.3.7.1. 7.2 def of Riemann integral#
partition of [a,b] is a finite set of points from [a,b] that includes both a and b
lower sum - sum all the possible smallest rectangles
a partition Q is a refinement of a partition P if \(P \subseteq Q\)
if \(P \subseteq Q\), then L(f,P)≤L(f,Q) and U(f,P)≥U(f,Q)
a bounded function f on the interval [a,b] is Riemann-integrable if U(f) = L(f) = \(\int_a^b f\)
iff \(\forall \epsilon >0\), there exists a partition P of [a,b] such that \(U(f,P)-L(f,P)<\epsilon\)
U(f) = inf{U(f,P)} for all possible partitions P
if f is continuous on [a,b] then it is integrable
3.3.7.2. 7.3 integrating functions with discontinuities#
if f:[a,b]->R is bounded and f is integrable on [c,b] for all c in (a,b), then f is integrable on [a,b]
3.3.7.3. 7.4 properties of Integral#
assume f: [a,b]->R is bounded and let c in (a,b). Then, f is integrable on [a,b] iff f is integrable on [a,c] and [c,b]. In this case we have \(\int_a^b f = \int_a^c f + \int_c^b f.\)F
integrable limit thm - Assume that \(f_n \to f\) uniformly on [a,b] and that each \(f_n\) is integarble. Then, f is integrable and \(lim_{n \to \infty} \int_a^b f_n = \int_a^b f\).
3.3.7.4. 7.5 fundamental theorem of calculus#
If f:[a,b] -> R is integrable, and F:[a,b]->R satisfies F’(x) = f(x) for all x \(\in\) [a,b], then \(\int_a^b f = F(b) - F(a)\)
Let f: [a,b]-> R be integrable and for x \(\in\) [a,b] define G(x) = \(\int_a^x g\). Then G is continuous on [a,b]. If g is continuous at some point \(c \in [a,b]\) then G is differentiable at c and G’(c) = g(c).
3.3.8. overview#
convergence
sequences
series
functional limits
normal, uniform
sequence of funcs
pointwise, uniform
series of funcs
pointwise, uniform
integrability
sequential criterion - usually good for proving discontinuous
limit points
functional limits
continuity
absence of uniform continuity
algebraic limit theorem ~ scalar multiplication, addition, multiplication, division
limit thm
sequences
series - can’t multiply / divide these
functional limits
continuity
differentiability
~integrability~
limit thms
continuous limit thm - Let (\(f_n\)) be a sequence of functions defined on A that converges uniformly on A to a function f. If each \(f_n\) is continuous at c in A, then f is continuous at c
differentiable limit theorem - let \(f_n \to f\) pointwise on the closed interval [a,b], and assume that each \(f_n\) is differentiable. If \((f'_n)\) converges uniformly on [a,b] to a function g, then the function f is differentiable and f’=g
convergent derivatives almost proves that \(f_n \to f\)
let (\(f_n\)) be a sequence of differentiable functions defined on the closed interval [a,b], and assume \((f'_n)\) converges uniformly to a function g on [a,b]. If there exists a point \(x_0 \in [a,b]\) for which \(f_n(x_0) \to f(x_0)\) is convergent, then (\(f_n\)) converges uniformly
integrable limit thm - Assume that \(f_n \to f\) uniformly on [a,b] and that each \(f_n\) is integarble. Then, f is integrable and \(lim_{n \to \infty} \int_a^b f_n = \int_a^b f\).
functions are continuous at isolated points, but limits don’t exist there
uniform continuity: minimize \(\|f(x)-f(y)\|\)
derivative doesn’t have to be continuous
integrable if finite amount of discontinuities and bounded