Proof without words 6

Il Teorema di Nicomaco

Image credits: https://en.m.wikipedia.org/wiki/Proof_without_words

Proofs without words 5

For every natural number $n$, the quantity $n^2-n$ is divisible by $2$.

Credits: Benjamin Dickman's answer in MO60457

Proofs without words 4

 Viviani's Theorem: In an equilateral triangle, the sum of the distances of an interior point to the three sides equals the altitude of the triangle.

Image from the web, attribution unknown

(Image from the web, attribution unknow

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