Suppose a sequence is defined by $\displaystyle a_0 = 0 $, $\displaystyle a_1 = 1 $, and $\displaystyle a_{n+1} = \frac{1}{2}\left(a_{n}+a_{n-1}\right) $ for $\displaystyle n \geq 2 $. Prove $\displaystyle a_n $ converges and determine its limit.

So $\displaystyle a_n = \frac{1}{2}\left(a_{n-1}+a_{n-2} \right) $. This means that for all $\displaystyle \varepsilon > 0 $, there exists $\displaystyle N \in \mathbb{N}$, such that whenever $\displaystyle n \geq N $, then $\displaystyle |a_{n}-L| < \varepsilon $ and $\displaystyle |a_{n+1}-L| < \varepsilon $.

Is this the right way to go about proving it?