How one could prove, that q pochhammer symbol $(1,1/n) = \prod_{k = 1}^{\infty}(1\frac{1}{n^k}) \geq 1  \frac{1}{n1}$
$$\prod (1x_i)=1x_1x_2(1x_1)x_3(1x_1)(1x_2)\ldots \geqslant 1\sum x_i, \forall x_i\in [0,1],$$ use this for $x_i=1/n^i$.
This function has an interesting asymptotic expansion. $$\prod_{k\ge1}(1n^{k}) = 1  n^{1}  n^{2} + n^{5} + n^{7}  n^{12}  n^{15} + \cdots\,\,. $$ The coefficients are all $\pm 1$ and the exponents are generalized pentagonal numbers.

$\begingroup$ Ah, by the way it also implies the inequality asked by OP! I remember that there was a question on MO with tremendously hard proofs of simple facts, this argument may be included in the collection. $\endgroup$ Apr 7 '19 at 20:57