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?
 A: Suppose for every $\epsilon >0$, there is an $N$ such that $n>N$ implies $x_n\leq\epsilon$. Let $k>0$. Then $\frac{k}{2}>0$, so there is an $M$ such that $n>M$ implies $x_n\leq\frac{k}{2}$ which implies that $x_n<k$. This corresponds to the original definition of continuity, doesn't it?
A: 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 $\eta>0$.
A: You are right, it doesn't matter.
Given an arbitrary $\epsilon > 0$, you can make your expression strictly less than $\epsilon$ by choosing your $\delta$ such that the expression is less than or equal to $\epsilon/2$, for example. (This is possible since you know that you can choose $\delta$ so that your expression is less than or equal to any positive number that you wish.)
A: There are already some good answers here, so I won't bother repeating them. However, I'd like to add a cautionary note that there are instances where using the wrong inequality gives nonsense — for example, it is essential in the statement of the Banach fixed point theorem that the Lipschitz constant is strictly less than one. If I remember correctly, even $\dfrac{\| f(x) - f(y) \|}{\| x - y \|} < 1$ for all $x \ne y$ is not good enough; we must have $\sup \left\{ \dfrac{\| f(x) - f(y) \|}{\| x - y \|} : x \ne y \right\} < 1$.
