Let $\displaystyle (x_{n})_{n}$ be a sequence of real numbers such that $\displaystyle x_{n} + x_{n+1}$ converges and $\displaystyle x_{n}x_{n+1}$ converges. Show that $\displaystyle (x_{2n})$converges
2. Well clearly then $\displaystyle a_n = x_{2n}+x_{2n+1}$ and $\displaystyle b_n = x_{2n}x_{2n+1}$ converge. Therefore $\displaystyle \frac{a_n\pm\sqrt{a_n^2-4b_n}}{2} = \{x_{2n},x_{2n+1}\}$ both converge.