An overkill proof of the divergence of the harmonic series

 The following proof can be found in the MO thread [1].

Assume that the harmonic series $\sum_{n=1}^{+ \infty}\frac{1}{n}$ converges. Then the sequence of functions $\{f_n\}$ defined by $f_n = \frac{1}{n} \chi_{[0, \, n]}$ is dominated by the function $$g = \chi_{[0, \, 1]} + \frac{1}{2} \chi_{[1, \, 2]}+\frac{1}{3} \chi_{[2, \, 3]}+ \frac{1}{4} \chi_{[3, \, 4]}+\ldots,$$ which is by assumption absolutely integrable over $\mathbb{R}$. 

By applying Lebesgue's Dominate Convergence Theorem [2] we get: $$1 = \lim_n \int_{\mathbb{R}} f_n(x) dx = \int_{\mathbb{R}}  \lim_n f_n(x)dx = 0,$$ a contradiction. $\Box$

References

[1] https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts

[2] https://en.wikipedia.org/wiki/Dominated_convergence_theorem