Expected number of cycles of permutation equals harmonic number of degree

Statement
Suppose $$n$$ is a natural number. Consider the uniform distribution on the symmetric group of degree $$n$$. Then, the expected number of cycles in the cycle decomposition of a permutation chosen according to the uniform distribution is equal to the harmonic number $$H_n$$ of $$n$$, where:

$$H_n := 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$$

Note that $$\lim_{n \to \infty} H_n - \log n = \gamma$$, where $$\gamma$$ is the Euler-Mascheroni constant, and its value is approximately $$0.577$$ (or very close to $$1/\sqrt{3}$$). Also, $$\lim_{n \to \infty} H_n/\log n = 1$$. Thus, for $$n$$ large enough, $$H_n$$ can be approximated additively as $$\log n$$ and multiplicatively as $$\log n$$.

See also probability distribution of number of cycles of permutation.