Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number α is irrational if and only if there are infinitely many rational numbers p/q, written in lowest terms, such that This was an improvement on Dirichlet's result which had 1/q2 on the right hand side.
The above result is best possible since the golden ratio φ is irrational but if we replace √5 by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ.
Again this new bound is best possible in the new setting, but this time the number √2 is the problem.
If we don't allow √2 then we can increase the number on the right hand side of the inequality from 2√2 to √221/5.
Repeating this process we get an infinite sequence of numbers √5, 2√2, √221/5, ... which converge to 3.