calculus - Limit of a Recursive Sequence

I'm having a really hard time finding the limit of a recursive sequence -

$$ \begin{align*}
&a(1)=2,\\
&a(2)=5,\\
&a(n+2)=\frac12 \cdot \big(a(n)+a(n+1)\big).
\end{align*}$$

I proved that the sequence is made up from a monotonically increasing sequence and a monotonically decreasing sequence, and I proved that the limits of the difference of these sequences is zero, so by Cantor's Lemma the above sequence does converge. I manually found out that it converges to $4$, but I can't seem to find any way to prove it.

Any help would be much appreciated!
Thank you.