# Math Help - Sequence proof

1. ## Sequence proof

I need help with the following proof:

Let $(x_{n})_{n}$ be a sequence of real numbers such that $x_{n} + x_{n+1}$ converges and $x_{n}x_{n+1}$ converges. Show that $(x_{2n})$converges

2. Well clearly then $a_n = x_{2n}+x_{2n+1}$ and $b_n = x_{2n}x_{2n+1}$ converge. Therefore $\frac{a_n\pm\sqrt{a_n^2-4b_n}}{2} = \{x_{2n},x_{2n+1}\}$ both converge.

3. why is it that we can just pass it though the quadratic formula?