Sequence of General Harmonic Numbers Converges for Index Greater than 1

Theorem
Let $H_n^{\paren r}$ denote the general harmonic number:
 * $\ds H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$

for $r \in \R_{>0}$.

Let $r > 1$.

Then as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.

Proof
For any $m \in \N$:

Since $m$ is arbitrary, every partial sum $H_n^{\paren r}$ is bounded from above by $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.

By Monotone Convergence Theorem, as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.