* __Theorem 13.5 (Bolzano-Weiestrass Theorem for Rn)__: Every bounded sequence (sm)m∈Rn has a convergent subsequence.
* __Theorem 13.5 (Bolzano-Weiestrass Theorem for Rn)__: Every bounded sequence (sm)m∈Rn has a convergent subsequence.
* __Topology__: Let S be a set. A topological structure on S is the data of a collection of subsets in S. This collection needs to satisfy:
* __Topology__: Let S be a set. A topological structure on S is the data of a collection of subsets in S. This collection needs to satisfy:
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- S and $\text{\o} are open
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- S and $\text{\o}$ are open
- arbitrary union of open subsets is still open
- arbitrary union of open subsets is still open
- finite intersections of open sets are open
- finite intersections of open sets are open
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* **Sequence and Convergence of Functions**:
* **Sequence and Convergence of Functions**:
* __Pointwise Convergence of Sequence of Sequences__: Let (xn)n be a sequence of sequences, xn∈RN, we say (xn)n converges to x∈RN pointwise if ∀i∈N, we have limn→∞xni=xi.
* __Pointwise Convergence of Sequence of Sequences__: Let (xn)n be a sequence of sequences, xn∈RN, we say (xn)n converges to x∈RN pointwise if ∀i∈N, we have limn→∞xni=xi.
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* Example: xni=n+ii, then this sequence of sequences converge to 0 pointwise, since for arbitrary fixed i, we have limn→∞xni=limn→∞n+ii=0.
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* Example: xni=n+ii, then this seq to 0 pointwise, since for arbitrary fixed i, we have limn→∞xni=limn→∞n+ii=0.
* __Uniform Convergence of Sequence of Sequences__: Let (xn)n be a sequence of sequences, xn∈RN, we say xn→x uniformly if ∀ϵ>0, ∃N>0 such that ∀n>N, $\sup\{\lvert x_{ni} - x_i \rvert :\, i\isin\natnums\} <\epsilon(alsoknownasd_{\infty}(x_n, x)$).
* __Uniform Convergence of Sequence of Sequences__: Let (xn)n be a sequence of sequences, xn∈RN, we say xn→x uniformly if ∀ϵ>0, ∃N>0 such that ∀n>N, $\sup\{\lvert x_{ni} - x_i \rvert :\, i\isin\natnums\} <\epsilon(alsoknownasd_{\infty}(x_n, x)$).
* Non-Example:xni=n+ii failed to converge uniformly to 0, since $d_{\infty}(x_n,0) = \sup \{\lvert x_{ni} \rvert :\, i\isin\natnums\} = \sup \{\frac{i}{n+i}:\, i\isin\natnums\} = 1$ (n is fixed).
* Non-Example:xni=n+ii failed to converge uniformly to 0, since $d_{\infty}(x_n,0) = \sup \{\lvert x_{ni} \rvert :\, i\isin\natnums\} = \sup \{\frac{i}{n+i}:\, i\isin\natnums\} = 1$ (n is fixed).
* __Pointwise Convergence of Sequence of Functions__: Given a sequence of functions fn∈ Map$(\Reals, \Reals),wesayf_nconvergetofpointwiseif\forall x\isin\Reals$,limn→∞fn(x)=f(x)⟺limn→∞∣fn(x)−f(x)∣=0.
* __Pointwise Convergence of Sequence of Functions__: Given a sequence of functions fn∈ Map$(\Reals, \Reals),wesayf_nconvergetofpointwiseif\forall x\isin\Reals$,limn→∞fn(x)=f(x)⟺limn→∞∣fn(x)−f(x)∣=0.
* __Rudin 7.9__: Suppose fn→f pointwise, then fn→f uniformly ⟺limn→∞(sup∣fn(x)−f(x)∣)=0.
* __Rudin 7.9__: Suppose fn→f pointwise, then fn→f uniformly ⟺limn→∞(sup∣fn(x)−f(x)∣)=0.
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* A sequence of functions {fn} is uniformly convergent to $f:D\to\Reals\iff \lim_{n\to\infty} \sup \{\lvert f_n(x) - f(x) \rvert : x\isin D\}$.
* __Rudin 7.10 (Weiestrass M-Test)__: Suppose f(x)=∑n=1∞fn(x)∀x∈X. If ∃Mn>0 such that supx∣fn(x)∣≤Mn and ∑nMn<∞, then the partial sum FN(x)=∑n=1Nfn(x) converges to f(x) uniformly.
* __Rudin 7.10 (Weiestrass M-Test)__: Suppose f(x)=∑n=1∞fn(x)∀x∈X. If ∃Mn>0 such that supx∣fn(x)∣≤Mn and ∑nMn<∞, then the partial sum FN(x)=∑n=1Nfn(x) converges to f(x) uniformly.
* Example: See last question on Midterm 2 version A.
* Example: See last question on Midterm 2 version A.
***Uniform Convergence and Continuity**:
***Uniform Convergence and Continuity**:
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* __Rudin 7.11__: Suppose fn→f uniformly on set E in a metric space. Let x be a limit point of E, and suppose that limt→xfn(t)=An. Then {An} converges and limt→xf(t)=limn→∞An. In conclusion, limt→xlimn→∞fn(t)=limn→∞limt→xfn(t).
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* __Rudin 7.11__: Suppose fn→f uniformly on set E in a metric space. of E, and suppose that limt→xfn(t)=An. Then {An} converges and limt→xf(t)=limn→∞An. In conclusion, limt→xlimn→∞fn(t)=limn→∞limt→xfn(t).
* __Rudin 7.12__: If {fn} is a sequence of continuous functions on E, and if fn→f uniformly on E, then f is continuous on E.
* __Rudin 7.12__: If {fn} is a sequence of continuous functions on E, and if fn→f uniformly on E, then f is continuous on E.
* __Rudin 7.13__: Suppose K compact and
* __Rudin 7.13__: Suppose K compact and
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* Example: [10,20]⊂R, then a partition would be P={10,15,18,19,20}.
* Example: [10,20]⊂R, then a partition would be P={10,15,18,19,20}.
* __U(P,f) and L(P,f)__: Given f:[a,b]→R bounded, and partion p={x0≤x1≤...≤xn}, we define U(P,f)=∑i=1nΔxiMi where $M_i= \sup \{f(x), x\isin [x_{i-1},x_i]\}$;L(P,f)=∑i=1nΔximi where $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.
* __U(P,f) and L(P,f)__: Given f:[a,b]→R bounded, and partion p={x0≤x1≤...≤xn}, we define U(P,f)=∑i=1nΔxiMi where $M_i= \sup \{f(x), x\isin [x_{i-1},x_i]\}$;L(P,f)=∑i=1nΔximi where $m_i= \inf \{f(x), x\isin [x_{i-1},x_i]\}$.
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{{ :math104:s:img_0a0c56a64ed8-1.jpeg?400 |}}
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{{ math104-s21:s:img_0a0c56a64ed8-1.jpeg?400 |}}
* __U(f) and L(f)__: Define U(f)=infPU(P,f) and L(f)=supPL(P,f).
* __U(f) and L(f)__: Define U(f)=infPU(P,f) and L(f)=supPL(P,f).
* Since f is bounded, hence ∃m,M∈R such that m≤f(x)≤M for all x∈[a,b], then ∀P partition of [a,b], U(P,f)≤∑i=1nΔxiM=M(b−a), and L(P,f)≥m(b−a), and m(b−a)≤L(P,f)leqU(P,f)≤M(b−a).
* Since f is bounded, hence ∃m,M∈R such that m≤f(x)≤M for all x∈[a,b], then ∀P partition of [a,b], U(P,f)≤∑i=1nΔxiM=M(b−a), and L(P,f)≥m(b−a), and m(b−a)≤L(P,f)leqU(P,f)≤M(b−a).
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*__Rudin 7.16__: Let α be monotone increasing on [a,b]. Suppose fn∈R(α), and fn→f uniformly on [a,b]. Then f is integrable and ∫abfdα=limn→∞∫abfndα.
*__Rudin 7.16__: Let α be monotone increasing on [a,b]. Suppose fn∈R(α), and fn→f uniformly on [a,b]. Then f is integrable and ∫abfdα=limn→∞∫abfndα.
*__Corollary__: Suppose fn∈R(α) and F(x)=∑n=1∞fn(x), the series converges uniformly, then F∈R(α) and ∫abF(x)dα=∑n=1∞∫abfn(x)dα.
*__Corollary__: Suppose fn∈R(α) and F(x)=∑n=1∞fn(x), the series converges uniformly, then F∈R(α) and ∫abF(x)dα=∑n=1∞∫abfn(x)dα.
- {{:math104:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but how to prove it converges to 5?
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- {{math104-s21:s:img_769dec51bb88-1.jpeg?400|}} I understand the visualization of this recursive sequence, but to 5?
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- In general, how to prove a set is infinite (in order to use theorem 11.2 in Ross)?
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- In general, how to prove a set is infinite(in order to use theorem 11.2 in Ross)?
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- Is there a way/analogy to understand/visualize the closure of a set? The definition is quite vague.
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- Is there a way/analogy to understand/visualize the closure of a set?
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- When should we use strong induction instead of regular induction? Will we get different results after using strong induction instead of induction?
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- Is there a way to actually test if a set is compact or not instead of merely finding finite subcovers for all open covers?
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- Is there a way to actually test if a set is compact or not instead of merely coming up with some open covers where the set is not finitely covered?
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- Rudin 4.6 states that if p∈E, a limit point, and f is continuous at p if and only if limx→pf(x)=f(p). Does this theorem hold for p∈E but not a limit point of E?
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- Rudin 4.6 states that if p∈Eas a limit point of E, then f is continuous at p if and only if limx→pf(x)=f(p). Does this theorem hold for p∈E but p is not a limit point of E?
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- {{math104-s21:s:img_0324.jpg?400|}} How is the claim at the bottom proved?
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- {{:math104:s:img_0324.jpg?400|}} How is the claim at the bottom proved?
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- Could we regard the global maximum as the maximum of all local minimums?
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- Could we regard global maxima as the maximum of all local maximums?
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- Under what circumstance would Taylor Theorem be essential to our proof, since for the question appeared in HW11 Q2, I really do not think using Taylor Theorem is a better way to prove the claim?
- Under what circumstance would Taylor Theorem be essential to our proof, since for the question appeared in HW11 Q2, I really do not think using Taylor Theorem is a better way to prove the claim?
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- In order for a Taylor series to converge (∑ncnzn), ∣z∣<R where R is the radium of convergence. But if ∣z∣=R, how can we determine whether the series is convergent or divergent?
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- In order for a Taylor series to converge (∑ncnzn), ∣z∣<R where R is the radium of convergence. But if ∣z∣=R, how can we tell?
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- If we are claiming f is continuous on $[a,b]$, how can we prove that f is continuous at the endpoints, i.e. do we just extend our interval to the left side of a and right side of b to do so?
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- If we are claiming f is continuous on [a,b], , i.e. do we just extend our interval to the left side of a and right side of b to do so?
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- What information could we extract from the line "f has a bounded first derivative (i.e. ∣f′∣≤M for some $M>0$)"?
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- What information can we extract from the line "f has a bounded first derivative (i.e. ∣f′∣≤M for some $M>0$)"?
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- How do we prove a set is sequentially compact without proving that it is compact? (Starting from its definition seems too complicated to take into account all sequences in the set)
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- How sequentially compact without proving that it is compact? (Starting from ms too complicated to take into account all sequences in the set)
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- If an+1=cos(an) and choose a1 such that 0<a1<1, is an a monotone sequence? Does it converge?
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- If an+1=cos(an) and choose a1 such that 0<a1<1, is an a ?
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- Does uniform convergence on a sequence of functions {fn} in F to f imply ?
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- If ∑fn converges uniformly, does it imply fn satisfies Weiestrass M-test?
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- For the alternating series test, if instead of sequence of numbers we have sequence of functions and those functions {fn} satisfies f1≥f2≥f3... and fn≥0 for all x∈X, limfn=0, does that mean ∑n(−1)nfn converges uniformly?
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- What is measure zero? (Related to Lebesgue measure and volume of open balls)
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- Is there any covering for a compact set that satisfies total length of the finite subcovers for the set is less than ∣b−a∣? (Answer is no)
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- Question 16 on Prof Fan's practice exam.
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- This is my solutions towards the practice exam: {{ math104-s21:s:practice_solutions.pdf |}}
math104-s21/s/martinzhai.1620488042.txt.gz · Last modified: 2021/05/08 08:34 by 173.205.94.2
