Prove $\lim b_{n}=0$

Question

$\lim_{n\to \infty}a_{n}b_{n}=0$ and $\lim_{n\to \infty}a_{n}=1$. Prove $\lim_{n\to \infty}b_{n}=0$

My approach: since $\lim_{n\to \infty}a_{n}b_{n}=0$ and $\lim_{n\to \infty}a_{n}=1$, by the limit laws of sequences, $\lim_{n\to \infty} \frac{a_{n}b_{n}}{a_{n}}=\frac{\lim_{n\to \infty}a_{n}b_{n}}{\lim_{n\to \infty}a_{n}}=\frac{0}{1}=0$. But since $0=\lim_{n\to \infty} \frac{a_{n}b_{n}}{a_{n}}=\lim_{n\to \infty} b_{n}$, we have $\lim_{n\to \infty} b_{n}=0$.

But what I want is an $\epsilon-\delta$ proof. Help?

Answer 1

Just do what you did before: $$ |b_n| = \frac{|a_n b_n|}{|a_n|} < \frac{\frac{1}{2}\varepsilon}{\frac{1}{2}} $$ provided that $|a_n| > 1/2$ and $|a_n b_n| < \frac{1}{2} \varepsilon$. But both conditions are surely true if $n$ is large enough, since $a_n \to 1 > 1/2$ and $a_n b_n \to 0$.