Replacing $\text{Expression}<\epsilon$  by $\text{Expression} \leq \epsilon$ As an exercise I was doing a proof about equicontinuity of a certain function. I noticed that I am always choosing the limits in a way that I finally get:
$\text{Expression} < \epsilon$
However it wouldn't hurt showing that 
$\text{Expression} \leq \epsilon$
would it, since $\epsilon$ is getting infinitesimally small? I have been doing this type of $\epsilon$ proofs quite some time now, but never asked myself that question. Am I allowed to write  $\text{Expression} \leq \epsilon$? If so, when?

Since $\varepsilon>0$ is arbitrary it does not matter. If you have a non strict inequality for each $\varepsilon>0$ you can get strict inequality by adding arbitrary $\eta>0$.
