It is required to prove that the recursive sequence $x_1 = 1$, $x_{n+1} = \frac{1}{2+x_n}$ is convergent.
Note that this sequence is oscillating, it converges to $\sqrt{2}-1$ and every element of the sequence is positive. Also note that $x_{2k}$ is an increasing sequence and that $x_{2k+1}$ is a decreasing sequence.
These are the correct facts that have been analyzed. However, it has not been possible to provide a proof of the convergence or of the above facts.
It is not clear how to prove that $x_{2k}$ is an increasing sequence, that is, in the induction hypothesis, it is given a $k \in \mathbb{N}$ such that $x_{2k} < x_{2(k+1)}$, which implies that $x_{2(k+1)+1} = \frac{1}{2+x_{2(k+1)}} < \frac{1}{2+x_{2k}} = x_{2k+1}$.
By a similar argument, since $x_{2(k+1)+1} < x_{2k+1}$, it follows that $$x_{2(k+1)} = \frac{1}{2+x_{2k+1}} < \frac{1}{2 + x_{2(k+1)+1}} = x_{2(k+2)}$$
Thus, $x_{2k}$ is an increasing sequence. By a similar argument, $x_{2k+1}$ is a decreasing sequence.
But only two subsequences and not any subsequence were taken in order to guarantee the convergence of the original sequence.
It this the correct and formal approach for showing that the sequence is convergent? Supposedly convergence can be proved without making use of Cauchy sequences and furthermore, proving convergence by the definition of epsilon is not immediate, so it has not been achieved.
Any valuable and kindly contribution is appreciated.
