Find the limit of $\frac{2+\sqrt{2U_n^2+4U_n+4}}{U_n}$
Picking up where you left off:
\begin{align} V_n=\frac{1}{2a_n+\sqrt{1+(2a_n+1)^2}} \end{align}
which means that $0<V_n<1$.
Now consider the two sub-sequences of $V_n$, which are $V_{2k}$ and $V_{2k+1}$, we have one of them must increase and the other must decrease.
Then by Weierstrass theorem, $V_{2k}$ and $V_{2k+1}$ converges.
Now it only remains to prove that $\lim V_{2k} = \lim V_{2k+1}$, but this would be straightforward since they have the same formula.
Hence, $\lim U_n = L$ exists. $\square$