Prove that $X_n \to c$ (in distribution) $⇒ X_n → c$ (in probability)

Question

Show that for real random variables $\{X_n\}$ on a probability space, $X_n \to c$ (in distribution) $⇒ X_n → c$ (in probability) ($c$ is the RV degenerate at 0)

My attempt:

I wrote down the definitions of the two convergence.

$P\{\omega : |X_n (\omega)-c(\omega)|>\epsilon\}=P\{\omega : |X_n (\omega)-c(\omega)|>\epsilon,\omega<c\}+P\{c : |X_n (c)-1|>\epsilon\}+P\{\omega : |X_n (\omega)-c(\omega)|>\epsilon,\omega>c\}$

But can't go further to show that the sum $\to 0$ as $n \to \infty$

Thanks in Advance for help!

Answer 1

$P\{X_n >c+\epsilon\} \to P\{c>c+\epsilon\}=0$ (by convergence in distribution) and (similarly) $P\{X_n <c-\epsilon\} \to P\{c<c-\epsilon\}=0$ .Then just add these two to get $P\{|X_n-c| >\epsilon \} \to 0$.