I am trying to find a sequence of rational numbers that converges to an irrational number. But the difficulty I am facing is that I am required to show convergence of such a sequence using only the $\varepsilon-N$ definition of sequences. The definition is:
We say $(a_n)\to a$ if for every $\varepsilon>0$, there is an $N\in\mathbb N$ such that $|a_n-a|<\varepsilon$ for all $n\ge N$.
I know that $(1+1/n)^n\to \mathrm e$ is a very popular example. But I cannot see how to prvoe the convergence using the above definition. Any help/hints would be appreciated.
P.S.: I checked out quite a few MSE questions regarding this but none of those examples seem to provable by the above definition.