Convergence in distribution to a constant implies convergence in probability

If $X_n\stackrel{d}{\to }c$ for fixed real $c$, convergence of the distribution functions at $c-\epsilon$ and $c+\epsilon$ shows $\mathbb{P}(|X_n-c|>\epsilon )\to 0$. Thus convergence in distribution to a deterministic limit implies convergence in probability to that limit, even though the implication fails for nonconstant limits on a specified probability space.