**Sketch of Apéry's Proof**

Apéry's first remark is that the double sequence

tends to uniformly in for in when tends to infinity. This stems from the alternating series being uniformly bounded by . However, the convergence of this series is not strong enough so as to show the irrationality of . Apéry used summation methods to accelerate the convergence. Namely, define

and ,

then also tends to . Here appears the crucial recurrence of Apéry: one remarks that it is satisfied by both sequences and , with initial conditions

, , and , .

By a number-theoretic argument, it follows from this recurrence that

,

with

and , which is positive.

This is sufficient to prove that is irrational, and yields an irrationality measure of at least