When proving convergence the part where $|a_n-a|<\varepsilon$, is it necessary to use a strict inequality? Could I use $\leq$ instead? Thanks.
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Yes we can use $\leq$ instead, but we usually don't because we set an arbitrary $\epsilon>0$. So if some quantity $M$ has the property that $ M \leq \epsilon$ for any $\epsilon > 0 $, then certainly if we replace the $\epsilon$ by $\epsilon/2 $, we see that we can also say that $ M \leq \epsilon/2 < \epsilon $ for any $\epsilon > 0 $. In the other way, if $M< \epsilon $ for any $\epsilon > 0$, then certainly $ M \leq \epsilon$ as well. Thus, allowing the equality is somewhat redundant and because we would rather be "cleaner" in our definitions, we exclude it.
No, it isn't. If for every $\epsilon$, you can find $N(\epsilon)$ such that $\forall n \ge N(\epsilon), |a_n - a| \le \epsilon$, then just pick an $\epsilon' < \epsilon$ and $N(\epsilon')$ will do the job.