Inequality for Sum of Square Roots Suppose $\{x_n\}_{n=1}^{\infty}$ and $\{y_n\}_{n=1}^{\infty}$ are two distinct sequences of positive numbers, each of which adds up to 1.  Is it necessarily the case that $\sum_{n=1}^{\infty}\sqrt{x_ny_n}<1$?

Yes, by AM-GM. Choose any index $k$ where $x_k \ne y_k$ and notice that
$$\sqrt{x_k y_k} < \frac{x_k + y_k}{2}$$
