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
