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Review

Real analysis​

Let A,A1,A2,…∈℘(X)A, A_1, A_2, \ldots\in \wp(\mathbb{X}). A1⊆A2⊆…A_1\subseteq A_2\subseteq\ldots and ⋃n∈NAn=A  ⟺  \bigcup_{n\in\N}A_n = A\iff the sequence (An)n∈N(A_n)_{n\in \N} increases to AA.

Similarly, A1⊇A2⊇…A_1\supseteq A_2\supseteq\ldots and ∩n∈NAn=A  ⟺  \cap_{n\in\N}A_n = A\iff the sequence (An)n∈N(A_n)_{n\in \N} decreases to AA.

∀x∈X,lim⁡n→∞1An(x)=1A(x)  ⟺  (An)n∈N\forall x\in \mathbb{X}, \lim_{n\to\infty}\mathbb{1}_{A_n}(x)=\mathbb{1}_A(x) \iff (A_n)_{n\in\N} converges to AA. lim⁡n→∞An=A\lim_{n\to \infty} A_n = A

  • lim⁡sup⁡n→∞An:=∩n∈N⋃k≥nAk\lim\sup_{n\to \infty} A_n := \cap_{n\in \N}\bigcup_{k\ge n} A_k and lim⁡inf⁡n→∞An:=⋃n∈N∩k≥nAk\lim\inf_{n\to \infty} A_n := \bigcup_{n\in \N}\cap_{k\ge n} A_k
  • Since ∀l,n∈N,∩k≥lAk⊆⋃k≥nAk\forall l,n\in \N, \cap_{k\ge l}A_k \subseteq \bigcup_{k\ge n} A_k, then lim⁡inf⁡n→∞An⊂lim⁡sup⁡n→∞An\lim\inf_{n\to \infty} A_n \subset\lim\sup_{n\to \infty} A_n

Formula and Functions reviews​

Binomial Formula: (a+b)n=∑k=0nnCk⋅akbn−k(a+b)^n = \sum\limits_{k=0}^n nCk\cdot a^kb^{n-k}

Multinomial Formula: (a1+a2+⋯+al)n=∑k1,k2,…,kln!k1!k2!…kl!∏i=1laiki(a_1 + a_2 + \cdots + a_l)^n = \sum_{k_1, k_2 , \ldots , k_l} \frac{n!}{k_1!k_2!\ldots k_l!} \prod_{i=1}^l a_i^{k_i}

The indicator function on A∈℘(X)A \in \wp(\mathbb{X}), denote as 1A:X→R\mathbb{1}_A :\mathbb{X}\to \R where 1A(x):={1x∈A0x∉A\mathbb{1}_A(x) := \begin{cases}1 & x\in A\\ 0 & x\notin A\end{cases}

  • 1A∩B=1A1B\mathbb{1}_{A\cap B} = \mathbb{1}_{A}\mathbb{1}_{B}
  • A∩B=∅  ⟹  1A∪B=1A+1BA\cap B = \empty \implies \mathbb{1}_{A\cup B} = \mathbb{1}_{A} + \mathbb{1}_{B}
  • 1A+1Ac=1\mathbb{1}_{A} + \mathbb{1}_{A^c} = 1