Products and Sums of Weighted Square-Summable Sequences of Real Numbers

In the setting of The Real Numbers: Standing Notation and Background, let $(a_{k})_{k\in\mathbb{N}}$, $(b_{k})_{k\in\mathbb{N}}$ and $(\mu_{k})_{k\in\mathbb{N}}$ be sequences of real numbers with $0\le\mu_{k}$ for every $k\in\mathbb{N}$, and let $\lambda\in\mathbb{R}$. Convergence of a series of real numbers and its sum are as defined there. Then the following hold.

1. (Pointwise bounds)(i) For all $s,t\in\mathbb{R}$,

$$2|st|\le s^{2}+t^{2}, \qquad (s+t)^{2}\le 2s^{2}+2t^{2}.$$

2. (Sums and multiples)(ii) Suppose the series $\sum_{k=1}^{\infty}\mu_{k}a_{k}^{2}$ and $\sum_{k=1}^{\infty}\mu_{k}b_{k}^{2}$ converge. Then the series $\sum_{k=1}^{\infty}\mu_{k}(a_{k}+b_{k})^{2}$ and $\sum_{k=1}^{\infty}\mu_{k}(\lambda a_{k})^{2}$ converge, and

$$\sum_{k=1}^{\infty}\mu_{k}(a_{k}+b_{k})^{2}\le 2\sum_{k=1}^{\infty}\mu_{k}a_{k}^{2}+2\sum_{k=1}^{\infty}\mu_{k}b_{k}^{2}, \qquad \sum_{k=1}^{\infty}\mu_{k}(\lambda a_{k})^{2}=\lambda^{2}\sum_{k=1}^{\infty}\mu_{k}a_{k}^{2}.$$

3. (Products)(iii) Suppose the series $\sum_{k=1}^{\infty}\mu_{k}a_{k}^{2}$ and $\sum_{k=1}^{\infty}\mu_{k}b_{k}^{2}$ converge. Then the series $\sum_{k=1}^{\infty}\mu_{k}|a_{k}b_{k}|$ and $\sum_{k=1}^{\infty}\mu_{k}a_{k}b_{k}$ converge, and

$$\Bigl|\sum_{k=1}^{\infty}\mu_{k}a_{k}b_{k}\Bigr| \le\sum_{k=1}^{\infty}\mu_{k}|a_{k}b_{k}| \le\tfrac{1}{2}\Bigl(\sum_{k=1}^{\infty}\mu_{k}a_{k}^{2}+\sum_{k=1}^{\infty}\mu_{k}b_{k}^{2}\Bigr).$$