Suppose I wanted to find the limit of some sequence, given by $\displaystyle a_{n+1} = \sqrt{2a_n}$. Suppose also that for this particular sequence, we have $\displaystyle a_n \leq a_{n+1} < 2$; i.e., the sequence in increasing and converges to 2. Rearranging the sequence would give $\displaystyle \frac{a_{n+1}^2}{a_n} = 2$. But $\displaystyle \displaystyle \lim _{n \to \infty} a_n = \lim _{n \to \infty} a_{n+1} = L$, and so $\displaystyle \displaystyle \lim _{n \to \infty} (a_n) = \lim _{n \to \infty} \frac{a_{n+1}^2}{a_n} = \frac{L^2}{L} = L = 2$.

I already know that this is the correct answer, but is there anything wrong with my methodology?