I have just learned about the convergence of a sequence with the epsilon definition.
So when we try to prove a limit of the sequence, what are we doing essentially (with respect to the definition)?
For example, if we want to show that $$\lim_{n\to\infty} 1/\sqrt{n} = 0,$$ then we want to show, by definition, for every $\varepsilon > 0$, there exists $M$ such that whenever $n\geq M$, $|a_n - 0| < \varepsilon$, in other words our $n^\text{th}$ term ($a_n$) should be less than $\varepsilon$. So we want to prove this last statement?
So is this correct: we want $a_n <\varepsilon$ to be true, so $1/\sqrt{n} <\varepsilon$, or $n > 1/\varepsilon^2$. In other words, we have that $M > 1/\varepsilon^2$ ? So to show this limit is indeed true, we want to be able to show the inequality is true or that we want to show we can always this $M$ by choosing its construction?