Tauber's theorem (Abel summable $\implies$ convergent) for $\sum c_n$ where $\lim_{n\to\infty} nc_n = 0$
Solution 1:
You've picked $N$ such that $n\geq N$ implies $n\vert c_n\vert <\epsilon$. This means
$$\begin{align}\left\vert\sum_{n=N}^{\infty}c_n\left(1-\frac{1}{N}\right)^{n}\right\vert &\leq \sum_{n=N}^{\infty}\vert c_n\vert\left(1-\frac{1}{N}\right)^{n} \\ &\leq \frac{\epsilon}{N}\sum_{n=N}^{\infty}\left(1-\frac{1}{N}\right)^{n}\\ &\leq \frac{\epsilon}{N}\frac{\left(1-\frac{1}{N}\right)^{N}}{1-(1-\frac{1}{N})}\\ &\leq \epsilon \left(1-\frac{1}{N}\right)^N \leq \frac{\epsilon}{e}\end{align}$$