This simple proof is well known ...
$$ 1 +1/2 +(1/3+1/4)+(1/5+1/6+1/7+1/8)+\cdots$$
Every sum in parentheses amounts to at least 1/2. If we take infinitely many sums, we get infinitely many times 1/2 or more, so that the total sum is not finite. Counting the pairs of parentheses, we find that not less than ¡0 of them are necessary7. Counting the fractions we find that Nicole de Oresme used 2¡0 natural numbers as denominators, not aware of Cantors celebrated theorem (1) according to which so many natural numbers are not available.