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- #1

Prove the Schwarz's and the triangle inequalities for infinite sequences:

If

$$

\sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad

\sum_{n = -\infty}^{\infty}|b_n|^2 < \infty

$$

then

$\displaystyle\left(\sum_{n = -\infty}^{\infty}|a_n + b_n|^2\right)^{1/2}\leq \left(\sum_{n = -\infty}^{\infty}|a_n|^2\right)^{1/2}\left(\sum_{n = -\infty}^{\infty}|b_n|^2\right)^{1/2}$.