The recursion formula for the sequence is $a_{n+1}=\frac{1}{2+a_{n}}, n \ge 1, a_{1}=\frac{1}{2}$. The hint of the question was to prove {$a_{2n+1}$} is decreasing and bounded below, and {$a_{2n}$} is increasing and bounded above. Therefore, according to MCT, the sequence converges, but I can't prove either them are monotonic increasing or decreasing. I know that the sequences converges to $\sqrt{2}-1$ and the recursion formula for the subsequences are $a_{2(n)+[1]}=\frac{2+a_{2(n-1)+[1]}}{5+2a_{2(n-1)+[1]}}$ but I don't know how to manipulate it rigorously as the fractional part is complicated.
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