Show that the sequence $y_n = 1+\frac{1}{2^k}+\frac{1}{3^k}+...+\frac{1}{n^k}$, where $n\in \mathbb{N}$ and $k\geq 2$ is convergent.
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$