Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares

In the setting of The Real Numbers: Standing Notation and Background, let $\alpha$ be a positive real number and let $M\in\mathbb{N}$. Convergence of a series of real numbers and its sum are as defined there. Then the following hold.

1. (Reciprocals reverse the order)¶ Let $a,b\in\mathbb{R}$ satisfy $0<a$ and $a\le b$. Then $a^{-1}$ and $b^{-1}$ exist, $0<b^{-1}$, and $b^{-1}\le a^{-1}$.

2. (Partial sums of the reciprocals of the squares)¶

3. (Summability)¶ The series $\sum_{m=1}^{\infty}\tfrac{1}{m^{2}}$ converges, and

4. (A quadratic sum about its centre)¶ Let $2M+1$ denote the natural number $M+M+1$. Then

each summand being defined because its denominator is positive.