Proof Regarding the Algebraic Limit Theorem

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Without using the property of the Algebraic Limit Theorem that states, $\lim\limits_{n \to \infty} a_n*b_n = ab$, Prove directly that $\frac{a_n}{b_n} \rightarrow \frac{a}{b}$, if $a_n \to a$ and $b_n \to b$ and b cannot equal zero.

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1 Answer

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Since, $b_n \to b$,given $\epsilon>0, $ $\exists N \in \mathbb{N}$, s.t. $\forall n\ge N, |b_n - b|<\epsilon$ and $|a_n-a|\le\epsilon$, then $|b_n|\ge\min \{|b_1|,\cdots,|b_N|,|b|-\epsilon\}=M(say)$ and $|a_n|\le\max \{|a_1|,\cdots,|a_N|,|a|+\epsilon\}=m(say)$.

$|\frac{a_n}{b_n}-\frac{a}{b}|=|\frac{a_n}{b_n}-\frac{a_n}{b}+\frac{a_n}{b}-\frac{a}{b}|\leq|a_n||\frac{1}{b_n}-\frac{1}{b}|+|1/b||a_n-a|\leq \frac{m\epsilon}{bM}+\frac{\epsilon}{b}=\frac{\epsilon}{b}(1+\frac{m}{M})$.

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