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)(i) 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)(ii)

$$\sum_{m=1}^{M}\frac{1}{m^{2}}\le 2-\frac{1}{M}.$$

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

$$\sum_{m=1}^{\infty}\frac{1}{m^{2}}\le 2 .$$

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

$$\sum_{m=1}^{2M+1}\frac{1}{1+\alpha\,(m-M-1)^{2}}\le 1+\frac{4}{\alpha},$$

each summand being defined because its denominator is positive.