Harmonic Series is Divergent/Proof 2

Theorem
The harmonic series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n$

diverges.

Proof
Observe that all the terms of the harmonic series are strictly positive.

To prove that the sequence is decreasing, consider:

$f: \R_{>0} \to \R$: $x \mapsto x^{-1}$

From the Power Rule for Derivatives:

Because $-x^{-2} < 0$ for all $x$ considered, from Derivative of Monotone Function, $f$ is decreasing.

As $f \left({n}\right)$ agrees with the harmonic series for all $n$ in the domain of $\displaystyle \sum \frac 1 n$, we conclude from Monotonicity of Real Sequences that the series is decreasing.

Hence the Cauchy Condensation Test can be applied, and we examine the convergence of:

This diverges, from the Nth Term Test.

Hence $\displaystyle \sum \frac 1 n$ also diverges.