I'm stuck in the following problem from Lima's "Análise Real":
Given $a > 0$, recursively define the sequence $(x_n)$ by letting $x_1 = 1 / a$ and $x_{n + 1} = 1/(a + x_n)$. Consider the real number $c > 0$ such that $c^2 + ac - 1 = 0$. Prove that $x_2 < x_4 < \dots < x_{2n} < \dots < c < \dots < x_{2n-1}< \dots < x_3 < x_1$, and that $\lim x_n = c$.
I understand that if I prove that the subsequence $(x_{2k})_{k \in \mathbb{N}}$ is increasing, with $c$ its least upper bound, then the result follows. I have tried manipulating the expressions to prove this monotonicity, but with no success. Any help would be much appreciated.