These two sequences have the same limit

These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and
$\{ b_n\}$ by
$$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$
$$b_n = \sqrt{a_{n-1}b_{n-1} }.$$
Prove that the sequences $\{a_n\}$ and $\{b_n\}$ converge and have the
same limit.
Source: Problem Solving Through Problems by Loren C. Larson.
Hint: