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Prove That...

Here are theorems and proofs that are essential in mathematical analysis. It would be nice if you could prove them smoothly.

Set

  1. (De Morgan's laws)
    • \((A \cup B)^C = A^C \cap B^C\)
    • \((A \cap B)^C = A^C \cup B^C\)
  2. Any countable union of countable sets is a countable set.

Limit of a sequence

  1. Prove with definition: \(\lim\limits_{n\to\infty}{\sqrt[n]{n}} = 1\).
  2. If \(\lim\limits_{n\to\infty}{a_n} = a\), then \(\lim\limits_{n\to\infty}{\dfrac{a_1+a_2+\cdots+a_n}{n}} = a\).
  3. For convergent sequences \(\{x_n\}\) and \(\{y_n\}\), if \(\lim\limits_{n\to\infty}x_n=a\), \(\lim\limits_{n\to\infty}y_n=b\), and \(a<b\), then exists \(N\in\mathbb{N}^\star\) so that \(x_n<y_n\) for all \(n>N\).
  4. If \(\lim\limits_{n\to\infty}x_n=a\), \(\lim\limits_{n\to\infty}y_n=b\neq0\), then \(\lim\limits_{n\to\infty}\dfrac{x_n}{y_n}=\dfrac{a}{b}\).
  5. \(a_n = \left(1 + \dfrac{1}{n}\right)^n\) and \(b_n = \left(1 + \dfrac{1}{n}\right)^{n + 1}\) converges to \(e\).
  6. \(a_n = 1 + \dfrac{1}{1!} + \dfrac{1}{2!} + \cdots + \dfrac{1}{n!}\) converges to \(e\).
  7. \(a_n = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n} - \ln n\) converges to \(\gamma\).
  8. \(\lim\limits_{n\to\infty}a_n=A\iff\lim\limits_{n\to\infty}a_{2n-1}=\lim\limits_{n\to\infty}a_{2n}=A\).
  9. \(\{a_n\}\) is convergent \(\iff\) every non-trivial subsequence \(\{a_{n_k}\}\) of \(\{a_n\}\) converges.
  10. \(a_n = \sin n\) is divergent.
  11. Prove with Nested interval theorem: \(\mathbb{R}\) is uncountable.
  12. Refer to Completeness of the Real Numbers, and prove
    1. Least upper bound \(\Rightarrow\) Monotone convergence.
    2. Monotone convergence \(\Rightarrow\) Nested interval.
    3. Nested interval \(\Rightarrow\) Bolzano-Weierstrass.
    4. Bolzano-Weierstrass \(\Rightarrow\) Cauchy criterion.
    5. Cauchy criterion \(\Rightarrow\) Least upper bound.
    6. Cauchy criterion \(\Rightarrow\) Nested interval.
    7. Nested interval \(\Rightarrow\) Least upper bound.

Limit of a function

  1. If \(\lim\limits_{x\to a}f(x)=A\), \(\lim\limits_{x\to a}g(x)=B\) and \(A>B\), then exists \(\delta>0\) so that \(f(x)>g(x)\) for all \(x\in\mathring{U}(x_0, \delta)\).
  2. If \(\lim\limits_{x\to x_0}f(x)=A\), then exists \(\delta>0\) so that \(f(x)\) is bounded in \(\mathring{U}(x_0, \delta)\).
  3. Prove with definition: \(\lim\limits_{x\to0}\dfrac{\sin x}{x} = 1\).
  4. (Heine's theorem) The necessary and sufficient condition for \(\lim\limits_{x\to a}f(x)=A\) is that for all sequences \(\{x_n\}\) which converges to \(a\) and \(x_n\neq a\), \(\lim\limits_{n\to\infty}f(x_n)=A\).
  5. Prove with Heine's theorem: \(f(x)=\sin\dfrac{1}{x}\) has no limit as \(x\to0\).
  6. (Cauchy criterion) Prove with Heine's theorem: \(\lim\limits_{x\to a}f(x)\) exists if and only if for all \(\varepsilon>0\), exists \(\delta>0\) so that \(|f(x)-f(y)|<\varepsilon\) for all \(x,y\in\mathring{U}(x_0, \delta)\).
  7. Prove with definition: \(\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=e\).

Continuity

  1. Every irrational point of \(R(x)\) is continuous, every rational point of \(R(x)\) is removable discontinuous, where
    \[R(x)=\left\{\begin{align}\frac{1}{q}&\quad \text{if}\ x=\frac{p}{q}\text{, with}\ p\in\mathbb{Z}\ \text{and}\ q\in\mathbb{N}\ \text{coprime.}\\0&\quad \text{if}\ x\ \text{is irrational.}\end{align}\right.\]
  2. If \(u=g(x)\) is continuous at \(x_0\), and \(y=f(u)\) is continuous at \(u_0=g(x_0)\), then \(f\circ g(x)=f(g(x))\) is continuous at \(x_0\).
  3. If \(f(x)\) is continuous in closed interval \([a, b]\),
    1. then it is bounded in \([a, b]\).
    2. then \(\max f(x)\) and \(\min f(x)\) exists in \([a, b]\).
    3. and \(f(a)\cdot f(b)<0\), then exists \(\xi\in(a, b)\) so that \(f(\xi)=0\).
    4. then it can reach all values between \(\min f(x)\) and \(\max f(x)\).
    5. (Cantor's theorem) then it is uniformly continuous in \([a, b]\).
  4. The sufficient and necessary condition for \(f(x)\) to be uniformly continuous in \(D\) is that for all sequences \(\{x_n\}, \{y_n\}\in D^\mathbb{N}\) that satisfies \(\lim\limits_{n\to\infty}(x_n-y_n)=0\), \(\lim\limits_{n\to\infty}(f(x_n)-f(y_n))=0\).
  5. If \(f(x)\) is continuous in finite open interval \((a, b)\), then \(f(x)\) is uniformly continuous on \((a, b)\) if and only if \(\lim\limits_{x\to a^+}f(x)\) and \(\lim\limits_{x\to b^-}f(x)\) exist.

Derivative

  1. (Darboux's theorem) If \(f(x)\) is differentiable in \((a, b)\), then for every \(y\) between \(f'(a)\) and \(f'(b)\), there exists \(\xi\in(a, b)\) so that \(f'(\xi)=y\).
  2. (Rolle's theorem) If \(f(x)\) is continuous in \([a, b]\), differentiable in \((a, b)\), and \(f(a)=f(b)\), then exists \(\xi\in(a, b)\) so that \(f'(\xi)=0\).
  3. If \(f(x)\) is twice differentiable in \([a, b]\), and \(f(a) = f(b) = 0\), then \(\forall x\in(a, b)\), \(\exists\xi\in(a, b)\), such that \(2f(x) = f''(\xi)(x - a)(x - b)\).
  4. If \(f'(x)\) is bounded in \((a, b)\), then \(f(x)\) is uniformly continuous in \((a, b)\).
  5. If \(f(x)\) is twice differentiable at \(x=0\), \(\lim\limits_{x\to0}\dfrac{f(x)}{x}=0\), and \(f''(0)=4\),
    1. find \(\lim\limits_{x\to0}\dfrac{f(x)}{x^2}\) and \(\lim\limits_{x\to0}\left(1+\dfrac{f(x)}{x}\right)^{1/x}\). (answer: \(2\) and \(e^2\))
    2. point out two mistakes: \(\lim\limits_{x\to0}\dfrac{f(x)}{x^2}=\lim\limits_{x\to0}\dfrac{f'(x)}{2x}=\lim\limits_{x\to0}\dfrac{f''(x)}{2}=\dfrac{1}{2}f''(0)=2\).
  6. Prove using Taylor series with Lagrange remainder: \(e\) is irrational.
Last update: 2023-11-24
Created: 2023-10-27