The sequence $\lbrace a _ n \rbrace$ is defined by $a_1 = 2$, and $$a_{n+1} = \dfrac{1}{2} \biggl( a_n + \dfrac{2}{a_{n}} \biggr),$$ for $n \geqslant 1$. Assuming that $\lbrace a_n \rbrace$ converges, find its limit.

$\lim \limits_{n\rightarrow\infty} a_{n} =$ .

Hint: Let $a = \lim \limits_{n\rightarrow\infty} a_ n$. Then, since $a_{n+1} = \frac{1}{2} \bigl( a_n + 2 / a_n \bigr)$, we have $a = \frac12 \bigl( a + 2/a \bigr)$. Now solve for $a$.