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)
5. 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.
6. 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):
1. Are sup and inf guaranteed to exist? If sup = infinity, do we say sup exists or not?
2. If Sn is bounded, is it guaranteed that limsup and liminf always exist?
3. Does pointwise convergence preserves continuity like uniform convergence does?
4. 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?
5. In hw11, problem 1, the reverse direction, h ⇐ f'(y) was left as an exercise. However I'm having trouble to prove this. Any hint?
6. Givan P1 = {1,4,6}, P2 = {1, 3, 3.5, 5, 5.6, 6}, is P2 a refinement of P1?
