Regarding the post below in :

There are just two things left I need to wrap my mind around, after that I think I will have comprehended the epsilon-delta concept.

In example 3 in the documentepsilon-delta1.pdfwhere the task is to show that $ \displaystyle \lim_{x \to 5} \, (x^2) = 25$, they assume that there exists an $ \displaystyle M$ such that $ \displaystyle |x + 5| \leqslant M$.

(1) Is it not supposed to be a strict inequality i.e. $ \displaystyle |x+5| < M$ and not $ \displaystyle |x+5| \leqslant M$? Why would the eventual equality between $ \displaystyle M$ and $ \displaystyle |x+5|$ ever be interesting?

They make the aforementioned requirement when one arrives at

$ \displaystyle |x-5| < \frac {\epsilon}{|x+5|} \, .$

We somehow, normally through algebraic manipulations, wish to arrive at $ \displaystyle |x-5| < \frac{\epsilon}{M}$ and in their procedure, they write

$ \displaystyle |x-5||x+5| < \epsilon \iff |x-5|M < \epsilon \, .$

(2) The steps above have overlooked something. Sure, I can buy that $ \displaystyle |x-5||x+5| < |x-5|M$ because we stipulated an upper bound for $ \displaystyle |x+5|$ but just because $ \displaystyle |x-5|M$ is greater than $ \displaystyle |x-5||x+5|$ does not mean that italsomust be less than epsilon, right?

Drawing a number line, one can readily conclude that having a < c and a < b does not imply b < c.

What is going on?