About some proofs using epsilon notation.

Question

There are many proofs in analysis where we find something like: "M=epsilon. This equality holds for any epsilon>0. So M=0.".

What is the background of this conclusion? Maybe I took the text out of context. I don't know.

For example this one:

Or this one:

Answer 1

it is more like "$M\leq\varepsilon$ for every $\varepsilon>0$". That is, for any $\varepsilon$ we have that $M$ is smaller than any positive real number. Now what is the largest nonnegative real number that is smaller than any $\varepsilon>0$? Note that it is important that $M$ is nonnegative. otherwise the reasoning would not be correct