## Question regarding above epsilon-delta proof

Hi guys. I have a question regarding the epsilon-delta proof you provided, on example 3.

Here's a snippet:

Example 3. ....... we will replace |x+5| by a number M which satisfies $\displaystyle |x+5| \leq M$. In so doing, we rewrite as $\displaystyle |x^2-25| < \epsilon \iff |x-5||x+5| < \epsilon \iff |x - 5|M < \epsilon \iff |x - 5| < \epsilon /M$
and proceed as before taking $\displaystyle \delta = \epsilon /M.$

My question is regarding the M. If M is $\displaystyle \geq$ |x+5|,
how do you conclude that |x+5||x-5| $\displaystyle < \epsilon \iff$ |x-5|M < $\displaystyle \epsilon$.

I would think the opposite. Since M is greater than |x+5|, then |x-5|M might actually be greater than epsilon!

For example: let's say x=9, epsilon = 58, M = 15

so we get |14||4| = 56 < 58,
however, |4| times 15 = 60 > 58!