Convergence of P-Series/Real/Proof 2

Proof
Let $p = 1$.

Then from Harmonic Series is Divergent the $p$-series diverges.

So let $p > 1$.

We note that the sequence of partial sums is increasing.

Hence it is sufficient to show that they are bounded above.

Let:
 * $s_{2^N} := 1 + \dfrac 1 {2^p} + \dfrac 1 {3^p} + \dotsb + \dfrac 1 {N^p}$

Then:

Hence the result.