An problem of irrational number approximation by rationals

Question

Suppose $\alpha$ is a positive irrational, and $\epsilon$ is an arbitrary positive real, are there $m,n$(non-negative integers) such that $$|\alpha-(2m+1)/(2n+1)|<\epsilon/(2n+1)?$$ If they exist, are there infinitely many $m,n$? It seems in this case the Dirichlet approximation theorem can not be used directly.