Okay so we all know the epsilon-N argument for convergence of sequences, that is a sequence $a_n$ converges to $a$ if $\forall \epsilon > 0, \exists N \in \mathbb{N} : n > N \implies |a_n - a| < \epsilon$
Now some point in my life, I've been told that any $\epsilon$ works, but I just cannot choose an $\epsilon$ that is dependent on $n$ because we would get a "changing epsilon".
So for instance, in proving the sum law for limits $\lim_{n\to \infty} a_n +b_n = L +M$ (provided the individual sequences' limits exists) we choose $\epsilon$ to be $\epsilon/2$ for for the partial sequences. But what happens if we choose $\epsilon/n$? So
$|a_n + b_n- L - M| \leq |a_n - L| + |b_n-M| < \frac{\epsilon}{2n}+ \frac{\epsilon}{2n} = \frac{\epsilon}{n}$. Okay so clearly $n$ is still positive, and I kinda see why writing $\epsilon$ in terms of $n$ here is dangerous, but when $n$ is big, can't still say $\epsilon/n < \epsilon$?