Basic Analysis: A sequence $b_n$ converges to nonzero $b$, prove that $\frac{1}{b_n}$ converges to $\frac{1}{b}$

Question

I've been struggling on this proof and I'd greatly appreciate a nudge in the right direction.

If a sequence $(b_{n})_{n\in\mathbb{N}}$ is never zero and converges to a non-zero term $b$, show that $\frac{1}{b_{n}}$ converges to $\frac{1}{b}$

I'm pretty sure I need to use triangle inequality and the definition of convergence, but beyond that I just can't put the pieces together. Thanks for reading

Answer 1

Hint : Estimate $|\frac{1}{b_n}-\frac{1}{b}|$ by $$ |\frac{1}{b_n}-\frac{1}{b}|=\frac{|b-b_n|}{|b.b_n|} $$ and "control" the denominator.