  Available proofs of result of the type 'at least one of the odd zeta values
$\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point
method or of linear independence criteria, or both. These two remarkable
techniques are however counted as highly non-elementary, therefore leaving the
partial irrationality result inaccessible to general mathematics audience in
all its glory. Here we modify the original construction of linear forms in odd
zeta values to produce, for the first time, an elementary proof of such a
result - a proof whose technical ingredients are limited to the prime number
theorem and Stirling's approximation formula for the factorial.
