# The Most Important Proofs in Prelims Analysis

> Source: https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/important-proofs/ · Updated: 2023-05-29 · Tags: uni, notes

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)
- [Course - Analysis II HT23](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/)
- [Course - Analysis III TT23](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/)

This is a completely objective list. My hope is that if I have these memorised, then I can sort of glue them together to work out other proofs.

### Analysis I
- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

#### Monotone sequence theorem
- [Notes - Analysis I MT22, Monotone sequence theorem](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/monotone-sequence-theorem/)
- Follows quite quickly from the suprema approximation property, the monotonicity of the sequence, and the definition of convergence

#### Scenic viewpoints theorem
- [Notes - Analysis I MT22, Scenic viewpoint theorem](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/notes/scenic-viewpoint-theorem/)
- Define a “scenic viewpoint set” which is either infinite or finite, in either case we get a monotone subsequence

#### Bolzano-Weierstrass theorem
- Follows directly from the scenic viewpoint theorem and the monotone sequence theorem put together.

#### Convergent iff Cauchy
- Sort of fiddly, for the Cauchy implies convergent direction you know the sequence is bounded, so has a monotone subsequence by Bolzano-Weierstrass. Then can mess around with the inequality.
- For the convergent implies Cauchy direction, you just do the “plus L, minus L” trick.

#### Limit form of comparison test
- First prove the normal form of the comparison test, which follows from $a_k$ being a monotone increasing sequence
- Then consider what $\frac{a_k}{b_k} \to L$ tells you by taking $\varepsilon = L/2$ and use the regular comparison test for both directions.

#### Alternating series test
- Group terms together in different ways in order to see that both the $s_{2n}$ and $s_{2n+1}$ series of partial sums are convergent by the monotone sequence theorem.

#### Ratio test
- Need to handle three cases seperately, but in both we compare with a geometric series. In the case where $0 \le L < 1$, this is $\alpha = \frac{1+L}{2}$ with $\varepsilon = \alpha - L$, in the case where $L > 1$, it’s also $\alpha = \frac{1+L}{2}$ but in the other case it’s $\alpha = 2$.

#### Integral test
- Show that $\sigma_n$ is bounded below and decreasing, by adding up all the inequalities like $f(n) \le \int^n_{n-1} f(x) \text d x \le f(n-1)$ to get $0 \le \sigma_n \le f(1)$ and then mess with $\sigma_{n+1} - \sigma_n$ to show it’s negative.
- Then the other result follows from the AOL.

#### Series converge only within their radius of convergence
- Two cases parts of the proof, absolute convergence within radius of convergence and then divergence outside it
- Absolute convergence within radius of convergence: There is some $S$ less than the radius of convergence, and let $\varepsilon = R - S$ to find a $\rho$ less than or equal to $R$ where the series converges absolutely. Then use the comparison test.
- Divergence outside radius of convergence: Use contradiction. Can bound the terms being summed by $M$, and then can find some $\rho$ where $R <  \rho < |z|$ where $0 \le |c_k \rho^k| \le |c_k z^k| \left| \frac \rho z \right|^k \le M \left |\frac \rho z\right|^k$ which converges as it’s a geometric series with common ratio less than $1$.

### Analysis II
- [Course - Analysis II HT23](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/)

#### Limit points via sequences
- [Notes - Analysis II HT23, Limit points](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/limit-points/)
- To get a sequence from the condition, take the $x_n$ given by letting $\varepsilon = \frac{1}{n}$.
- To get the condition from the sequence, let $\varepsilon$ be arbitrary, and then $\exists p_N$ such that $0 < |p_N - p| < \varepsilon$.

#### Function limits via sequences
- [Notes - Analysis II HT23, Function limits](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/function-limits/)

#### AOL for functions
- [Notes - Analysis II HT23, Function limits](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/function-limits/)

#### Limits of composition of functions
- [Notes - Analysis II HT23, Function limits](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/function-limits/)

#### Boundedness theorem
- [Notes - Analysis II HT23, Boundedness theorem](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/boundedness-theorem/)

#### Intermediate value theorem
- [Notes - Analysis II HT23, Intermediate value theorem](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/intermediate-value-theorem/)

#### Continuous inverse function theorem
- [Notes - Analysis II HT23, Continuous inverse function theorem](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/continuous-inverse-function-theorem/)

#### Continuity on closed bounded intervals implies uniform continuity
- [Notes - Analysis II HT23, Uniform continuity](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/uniform-continuity/)

#### Uniform limit of continuous functions is continuous
- [Notes - Analysis II HT23, Uniform convergence](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/uniform-convergence/)

#### Weierstrass’ M-test
- [Notes - Analysis II HT23, Uniform convergence](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/uniform-convergence/)

#### Uniform convergence of power series within their radius of convergence
- [Notes - Analysis II HT23, Uniform convergence](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/uniform-convergence/)

#### Equivalence of two definitions of differentiation
- [Notes - Analysis II HT23, Differentitation](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/differentitation/)

#### Fermat’s theorem
- [Notes - Analysis II HT23, Extrema](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/extrema/)

#### Rolle’s theorem
- [Notes - Analysis II HT23, Mean value theorems](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/mean-value-theorems/)

#### (Generalised) Mean value theorem
- [Notes - Analysis II HT23, Mean value theorems](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/mean-value-theorems/)

#### Taylor’s theorem
- [Notes - Analysis II HT23, Taylor’s theorem](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/taylors-theorem/)
- Analyse the remainder term, and can show by induction with the generalised mean value theorem that this satisfies a property that implies the overall theorem.

#### L’Hopital’s Rule
- [Notes - Analysis II HT23, L’Hôpital’s Rule](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/lhopitals-rule/)
- Proofs in the lecture notes aren’t very detailed
- $\frac 0 0$ case invovles showing that $g(x) = g(x) - g(a)$ is nonzero and then using Cauchy’s mean value theorem.
- $\frac \infty \infty$ case involves some very dodgy steps that I can’t really justify, but all stems from the fact that for some reason, you can find a $\delta' \in (0, \delta)$ such that $\left|\frac{f(x) - f(c)}{g(x) - g(c)} - L\right| < \varepsilon$. Then some rearranging (which might actually be wrong in the lecture notes, or at least misleading) gives the required result. 

#### Real binomial theorem
- [Notes - Analysis II HT23, Binomial expansion](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/binomial-expansion/)
- Show that the quotient $F(x) = \frac{\sum^\infty_{k=0} {p \choose k} x^k}{(1+x)^p}$ has zero derivative and so this function is constant, using the fact that for both $f(x) = (1+x)^p$ and $g(x) = \sum^\infty_{k=0} {p \choose k} x^k$, we have $(1+x)f'(x) = pf(x)$.

### Analysis III
#### Definition of integrability in terms of epsilon
- [Notes - Analysis III TT23, Step functions and basic definitions](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/step-functions-and-basic-definitions/)
- Forward direction follows from the supremum and infimum approximation property
- Backwards direction follows from considering the supremum and infimum as bounds and then applying this to $I(\phi_+) - I(\phi_-) < \varepsilon$ to show that they can be squeezed arbitrarily close together.

#### If integrable on an interval, integrable on parts of that interval
- [Notes - Analysis III TT23, Basic theorems about the integral](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/basic-theorems-about-the-integral/)
- Consider that a majorant for the interval is the same as two “sub-majorants” juxtaposed next to each other (under some assumptions that we can take without loss of generality)
- Use the fact that $x + y = x' + y'$ and $x \le x'$ and $y \le y'$ implies $x = x'$ and $y = y'$ applied to the supremums and infimums.

#### Linearity of integration
- [Notes - Analysis III TT23, Basic theorems about the integral](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/basic-theorems-about-the-integral/)
- Use the $\varepsilon$ charaterisation of integration, just consider $\phi_- + \psi_-$ and $\phi_+ + \psi_+$ and exploit the fact that $I$ is linear.

#### Any continuous function is integrable (on a closed interval)
- [Notes - Analysis III TT23, Basic theorems about the integral](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/basic-theorems-about-the-integral/)
- Heavily uses the fact that continuity on a closed interval implies uniform continuity.
- Consider a partition with mesh less than the $\delta$ from uniform continuity, and then the optimal majorants and minorants on that interval, to eventually show that $I(\phi_+) - I(\phi_-)$ is bounded by $\varepsilon(b - a)$. 

#### First fundamental theorem of calculus
- [Notes - Analysis III TT23, Fundamental theorems of calculus](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/fundamental-theorems-of-calculus/)
- Actually can show it’s Lipschitz, you just consider $|F(c+h) - F(c)|$ and then you know that this translates to the integral of a bounded function.
- Showing that if $f$ is continuous then $F$ is differentiable involves considering $|F(c+h) - F(c) - hf(c)|$ which translates to $\left|\int^{c+h}_c (f(x) - f(c)) \text d x\right|$.

#### Second fundamental theorem of calculus
- [Notes - Analysis III TT23, Fundamental theorems of calculus](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/fundamental-theorems-of-calculus/)
- Use Riemann sums and the mean value theorem applied to $F'$.
- Given all the conditions, for any $(x_{i-1}, x_i)$, $\exists \xi$ s.t. $F'(\xi) = \frac{F(x_i) - F(x_{i-1})}{x_i - x_{i-1}}$.
- This sum telescopes to the required value.

#### Integration by substitution
- [Notes - Analysis III TT23, Integration by substitution](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/integration-by-substitution/)
- Both these integration technique proofs use the second fundamental theorem of calculus.
- First note that the whole thing is actually integrable.
- Then consider $F(x) = \int^x_a f$ and then think about $(F \circ \phi)'$.

#### Integration by parts
- [Notes - Analysis III TT23, Integration by parts](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/integration-by-parts/)
- Consider $F = fg$ and then just use the second fundamental theorem of calculus.

#### Integration and uniform limits commute ($\ast\ast$)
- [Notes - Analysis III TT23, Integration and limits](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/integration-and-limits/)
- You know there exists some $f_n$ where $|f_n - f|$ is less than $\varepsilon$ everywhere and also have majorants and minorants for $f_n$. So you can define $\hat\phi_-$ and $\hat \phi_+$ by adding and subtracting $\varepsilon$ respectively, which will define new majorants and minorants.
- Then note $I(\hat \phi_+) - I(\hat \phi_-)$ is some constant multiple of $\varepsilon$
- To show the integrals are actually equal, consider $|\int^b_a (f_n - f)|$.

#### Differentiation and limits commute, sometimes ($\ast\ast$)
- [Notes - Analysis III TT23, Differentiation and limits](https://ollybritton.com/notes/uni/prelims/tt23/analysis-iii/notes/differentiation-and-limits/)
- The limit $f'_n \to g$ needs to be uniform and $g$ needs to be bounded because you need to show $g$ is integrable.
- Define $F(x) = \int^x_a g$, so that $F' = g$
- Consider $\int^x_a f'_n(x) = f_n(x) - f_n(a)$ and then take the limit on both sides
- End up with equality with $F$.

#### Differentiation theorem for power series
- [Notes - Analysis II HT23, Differentiation theorem](https://ollybritton.com/notes/uni/prelims/ht23/analysis-ii/notes/differentiation-theorem/)
- Relies on two results, that $\sum^\infty_{i=0} \lambda^i$ and $\sum^\infty_{i=1} i\lambda^{i-1}$ both converge for $|\lambda|<1$
- Then you use the differentiation and limits commuting result for series, and check the uniform convergence via the M-test.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt