Given two sequence $\{x_{k}\}_{k\ge 1}, \{y_{k}\}_{k\ge 1},$ Holder's inequality states that
$$\sum_{k=1}^{\infty} |x_k\,y_k| \leq \biggl( \sum_{k=1}^{\infty} |x_k|^2 \biggr)^{\frac{1}{2}} \biggl( \sum_{k=1}^{\infty} |y_k|^2 \biggr)^{\frac{1}{2}}
$$
How we can verify that
$$\left(\sum_{k\ne l} |x_k|^2\,|x_l|^2 \right)^\frac{1}{2}\leq \sum_{k=1}^{\infty} |x_k|^2
$$
Best Answer
$\sum_{k \neq l}|x_k|^{2}|x_l|^{2} \leq \sum_{k,l}|x_k|^{2}|x_l|^{2} =(\sum|x_k|^{2})^{2}$ Take square root on both sides.
[I have used the fact that $(\sum_k a_k)^{2} =\sum_{k,l} a_ka_l$].