$\displaystyle show \ that \ the \ sequence :$
$\displaystyle x_1=3 \ , \ x_{n+1}= \frac{x_n}{2}+ \frac{4}{x_n}$
$\displaystyle is \ convergent \ and \ find \ its \ limit?$
The same, prove that is a monotone bounded sequence.
Besides $\displaystyle x_n$ is a subsequence of $\displaystyle x_{n+1},$ so both sequences converge to the same limit (which you need to prove that exists, so call it $\displaystyle l$), then the limit is $\displaystyle l=\frac l2+\frac4l.$