 Solution

Fix $\delta >0$. Once $|c_n-c|<\delta$, the following event inclusions hold outside $\{|X_n-X|>\delta \}$:Taking probabilities gives the explicit boundsBy the assumed convergence in probability, the error tends to zero. ThereforeThe continuity of the cumulative distribution function $F$ at $c$ lets $\delta$ decrease to zero, proving $\boxed{F_n(c_n)\to F(c).}$ This proves the needed implication directly, without invoking any theorem comparing convergence of random variables.