Convergence in distribution to a constant implies convergence in probability
= Convergence in distribution to a constant implies convergence in probability
If $X_n\xrightarrow d c$ for fixed real $c$, convergence of the <distribution functions> at $c-\varepsilon$ and $c+\varepsilon$ shows $\mathbb P(|X_n-c|>\varepsilon)\to0$. 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>.