Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$ So I am stuck on this question and am unsure how to solve it, would be appreciated if someone could help me out, thanks! :)
Question
Consider a sequence $(b_n)_{n=1}^{\infty}$ of non-zero real numbers. By definition, the infinite product $\prod_{n=1}^{\infty}$ converges if the sequence $(p_n)_{n=1}^{\infty}$, where
$p_n=\prod_{k=1}^{n}b_k$
converges to some non-zero number. Prove that the convergence of $\prod_{n=1}^{\infty}b_n$ implies $\lim_{n\to\infty}b_n=1$

Hint:
$$b_n = \frac{\prod_{k=1}^nb_k}{\prod_{k=1}^{n-1}b_k}$$
