Sequence of General Harmonic Numbers Converges for Index Greater than 1/Proof

Proof
Let $m \in \N$ be arbitrary.

Then:

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