proving $\lim_{n\to\infty} |b_n| =1$

Question

How do I prove that given $\lim\limits_{n\to \infty}\left(a_n\cdot b_n\right)=1$, that if $\lim\limits_{n\to \infty}\left(|a_n|\right)=1$, then $\lim\limits_{n\to \infty}\left(|b_n|\right)=1$? (I tried to say that $\lim\limits_{n\to \infty}\left(|b_n|\right)$ is not 1, but got stuck; and also tried the direct proof by taking $\varepsilon/2$ for both but it didn't turn out well). Appreciated if someone can help.

Answer 1

first since $$\lim_{n \rightarrow \infty} a_n.b_n =1 \implies \lim_{n \rightarrow \infty}|a_n.b_n|= |1|$$

and since $\lim_{n \rightarrow \infty} |a_n| = 1 $ after certain point it will be nonzero. $$\lim_{n \rightarrow \infty} |a_n| = 1 \implies \exists N \,\,\text{s.t} \,\, ||a_n|-1| < 1/2 \implies |a_n| \in(1/2,3/2)$$

we will use following statement

$$\lim_{n \rightarrow \infty} p_n = L , \lim_{n \rightarrow \infty} q_n = M (\neq 0) \implies \lim_{n \rightarrow \infty} \Big(\frac{p_n}{q_n} \Big) = \frac{L}{M}$$ now $$|b_{N}| = \Big|\frac{a_N.b_N}{a_N}\Big| \implies \lim_{N \rightarrow \infty} |b_N| = \frac{\lim_{N \rightarrow \infty}|a_N.b_N|}{\lim_{N \rightarrow \infty} |a_N|} = \frac{1}{1} = 1$$