proof of ratio test

Question

I don't understand two parts in this proof.

1. At first, it uses $n \ge M$, and later uses $n \ge M+1$ Is there any reason to change weak to strong inequality??

2. If $|x_n|$ is replaced by $|x_M|r^{-M}r^n$, shouldn't it be a strong inequality?

Answer 1

For first question: If $n>M$, $n\ge M$ is also true, so we can use the result from before after restricting ourselves to 'strong inequality'.

Second question: If $a<b$, $a\le b$ is also true, so the step is valid . Of course you can write the step $\dots \lt \dots$, 'strong inequality', but that is just uneccesary.