This is an old revision of the document!
I have kept most of my notes handwritten throughout the semester. Instead of taking tons of photos of my handwritten notes, I will pick some of the important concepts and theorems to put on this website so that I can review the final in a more efficient way. Feel free to comment on my website:)
1. Number System:
Natural numbers:
- allowed operations: +, *
- exists successor
- 1 belongs to N, n belongs to N, n+1 belongs to N
- basis of induction
Integers:
- distinguished 0
Rationals:
- example of field
- ordered field
Completeness axiom:
- If S is bounded above, then sup(S) exists. If also bounded below, inf(S) = -sup(-S)
Archimedian property:
- if a,b>0, then there exists a natural number n such that n*a>b. Prove by contradiction
Rational roots theorem: - If r = c/d != 0 a rational number, and r satisfies cn*x^n + cn-s*x^n-1 +…+c0=0 with ci in Z, cn, c0 not equal to 0, then d|Cn, and c|C0
2. Sequences:
Sequence and limit:
- use epsilon delta proof
- Remember cauchy definition implies convergence for reals (or complete metric space)
Note: all convergent sequences are bounded.
Useful results of limits: preserves addition, subtraction, multiplication, and division when denominator not equal to 0.
Note:
- let An be bounded, then An converges iff limsup(an) = liminf(an)
- for all epsilon, there is a natural number N>0 such that for all n>N, an<limsup(an)+epsilon
Bounded monotone sequence: - Bounded monotone sequences converge.
Subsequence:
Thm: let Sn be s sequence, and a real number t, then (Sn) has a subsequent converges to t iff for all epsilon>0, the set Aepsilon{natural number n| abs(Sn-t) < epsilon} is infinite.
Note:
- Every sequence has a monotone subsequence.
- Therefore, every bounded sequence has a convergent subsequence(bolzano-welstress)
Subsequential limit:
- The limsup(Sn) and liminf(Sn) are subsequential limits of Sn. And sup(S*) = limsup(Sn), inf(S*) = liminf(Sn) where S* is the set of all subsequential limits of Sn. If S* has a single element, then all subsequent converge to lim(Sn)
- The S* for bounded Sn is closed.
Important inequality given nonzero real sequence an:
$\liminf |a_{n+1}/a_n| \leq \liminf |a_n|^\frac{1}{n} \leq \limsup |a_n|^\frac{1}{n} \leq \limsup |a_{n+1}/a_n|$
Technique:
- When proving Sn diverges to infinity, try to find for all M>0 there exists N>0 such that for all n>N, Sn>M
- Triangle inequality
3. Metric Space
Properties of distance metrics:
- d(x,y) >= 0 and only equal to zero when x=y
- d(x,y) = d(y, x)
- d(x,y) + d(y,z) >= d(x,z)
Note:
Convergence of Sn implies cauchy, but the converse is true only when the metric space is complete.
- A metric space is complete if every cauchy sequence in it has a limit.
Open and close:
- finite intersection of open set is open
- arbitrary union of open set is open
- finite union of closed set is closed
- arbitrary intersection of closed set is closed
Closure property:
- If x in the closure of the set S, then for all epsilon>0, epsilon ball(x) intersects with E is not empty.
Heine-Borel:
- Let K be a subset of Rn, then K compact iff K is closed and bounded.
note(converse is true only in Rn)
Sequential compactness:
- S is sequential compact if for all infinite sequence Sn in S has a convergent subsequence. Note: Sequential compact ⇔ covering compact
Interesting observations:
- the boundary of a ball is not always a sphere (consider a discrete metric).
- closure of Q is R
- with L1 metric, an open ball is a diamond
- with L infinity metric, an open ball is a square
4. Series:
Note:
- The cauchy of a series implies it converges.
Useful formula:
- 1+x+x^2+…+x^n = (1-x^(n+1))/(1-x)
- infinity summation of a*r^n starting from index 0 = a/(1-r) where |r| < 1
Series tests:
- comparison test
- root test - absolute convergence
- ratio test - absolute convergence
- alternating series test
- integral test (continuous, positive, decreasing)
6. Functions between metric spaces:
Reminder:
- Notice the difference between uniform convergence and convergence on a single point.
Different definitions of continuity:
- A function F from X to Y is continuous iff for all V in Y open, F^-1(V) is open.
- A function F from X to Y is continuous iff for all p in X' we have f(p) = lim f(x):x →p
Related theorem:
- let F:X→Rn, with F(x) = (f1(x), f2(x),…,fn(x)) then f is continuous ⇔ fi:x→R are continuous for all
i = 1,…,n
induced topology:
- let X be a topological space, S is a subset of X, we can equip S with induced topology E within S is open in S iff there exists open subset E2 such that E = intersection of S and E2.
Thm:
- f:X → Y continuous, then if E within X, then f(E) is compact in Y
- f:X → Y continuous, and X is compact, then f is uniformly continuous.
- f:X → Y continuous, and X is compact, and f is a bijection, then f inverse if continuous.
Connectedness:
Let X be a topological space:
- X is connected iff the only subset of X that is both open and closed are X and empty set.
- X is not connected iff there exists U, V in X nonempty, open, and disjoint such that X = U union V.
Thm:
- f:X → Y continuous, and X is connected, then f(X) is connected.
- IVT
- MVT
Note:
F is continuous at p iff f(p) = f(p+) = f(p-)
- if f(p+) and f(p-) exist, but not continuous at p, then first kind discontiuity
- otherwise, second kind discontinuity.
Thm:
- if f is monotone increasing on (a,b), then f(x+) and f(x-) exists at every point in (a,b) and has at most countably many discontinuities which are all first kind.
Convergence of functions:
- pointwise convergence vs. uniform convergence
- bump function as example
- pointwise convergence does not preserve integration nor differentiation
Uniform convergence:
- l infinity metric: d(fn, f) = 0 as n goes to infinity.
- uniform cauchy implies uniform convergence
- preserves continuity
Thm:
Let K be compact metric space, fn:K→R
1. fn is continuous for all n;
2. fn(x)→f(x) and f is continuous for all x
3. fn(x)>=fn+1(x) for all x and n
then fn converges to f uniformlly.
5. Differentiability and Integrability
Prop:
- Differentiability implies continuity
- Differentiability ⇒ there exists u(x) such that f(x) = f(p)+(x-p)f'(p) + (x-p)u(x) where lim u(x) as x goes to p = 0
Caveat:
- f' exists everywhere doesn't imply that f' is continuous.
Rolle's theorem
- if f continuous on [a,b] and f' exists for all x in (a,b) if f(a) = f(b) then there exists c in (a,b) such that f'© = 0
Generalized Mean Value Theorem
- if f, g continuous and differentiable on (a,b), then there exists c in (1,b) such that [f(b) - f(a)]g'© = [g(b) - g(a)]f'\©
Corollary:
- if f differentiable and bounded by M>0, then f is uniformally continuous.
L'Hopital Rule
- lim f(x)/g(x) = lim f'(x)/g'(x) if lim f(x)/g(x) = 0/0 or inf/inf
Taylor's Theorem
- given f that is smooth till n-1 degree, and is continuous on on interval $[a, b]$, and given $\alpha, \beta \in [a, b]$, we have $$f(\beta) = \sum_{i=0}^{n-1}\frac{f^{(i)}(\alpha) (\beta - \alpha)^i}{i !} + \frac{f^{(n)}(\gamma)(\beta - \alpha)^n}{n!}$$ for some $\gamma$ between $\alpha$ and $\beta$.
Riemman Integral
Idea:
- A partition P of interval [a,b] is a = x0<x1<…xn=b.
- Define U(P,f) and L(P,f)
- Let U(f) = inf U(P,f) and L(f) = sup L(P,f)
- f is integrabvle if U(f) = L(f)
Generalization:
- Let alpha be a monotone increasing function, define interval delta alpha = alpha(xi) - alpha(xi-1)
- If U(P, alpha) = L(P, alpha), then it is riemman-stieltjes integrable with respect to alpha.
Refinement lemma:
- If Q is a refinement of P on [a,b],then the approximated integral bounds get better: Lp ⇐ Lq ⇐ Uq ⇐ Up
Cauchy condition of integral:
- f is integrable w.r.t. alpha iff for all epsilon > 0, there exists p partition such that Up - Lp < epsilon.
Related theorems:
- If f is continuous on [a,b], then f is integrable w.r.t. alpha on [a,b]
- If f is monotonic, and alpha continuous, then f is integrable
- If f is discontinuous only at finitely many points and alpha is continuous where f is discontinuous, then f is integrable
- if f:[a,b]→[m,M] and phi:[m,M]→R is continuous, if f is integrable wrt alpha, then h = phi composed with f is integrable wrt alpha
- integration operation is linear in both f and alpha
- composition of integrable functions is integrable
- |f| is integrable if f is integrable
- let alpha be monotone increasing and alpha' exists and is riemman integrable wrt to dx, then bounded f integrable ⇔ f*alpha'is integrable
- change of variable
- integration by parts
- uniform convergence preserves integration
Questions (updating):
* Are sup and inf guaranteed to exist? If sup = infinity, do we say sup exists or not?
* If Sn is bounded, is it guaranteed that limsup and liminf always exist?
* Does pointwise convergence preserves continuity like uniform convergence does?
* Extra question from hw:Let $f(x)$ be a differentiable function on $[-1,1]$ with $f'(x)$ continuous. Assume $x=0$ is the unique global minimum of $f$, i.e., for any $x \neq 0$, we have $f(x) > f(0)$. Is it true that there exists a $\delta > 0$, such that $f'(x) < 0$ for $x \in (-\delta, 0)$ and $f'(x)>0$ for $x \in (0 ,\delta)$? (Just as the case if $f(x)=x^2$) –I think should be true?
* I’m having a hard time to imagine why why would 1/x be integrable on R? Since it is discontinuous only at x=0, and f(x) = x is continuous at x=0, it is integrable. But what about the strange behavior near x=0?
