= Solution
{c}
Fix $\delta>0$. Once $|c_n-c|<\delta$, the following event inclusions hold outside $\{|X_n-X|>\delta\}$:
$$
\{X\leq c-2\delta\}\subseteq\{X_n\leq c_n\}\subseteq\{X\leq c+2\delta\}.
$$
Taking <probabilities> gives the explicit bounds
$$
F(c-2\delta)-\mathbb P(|X_n-X|>\delta)\leq F_n(c_n)\leq F(c+2\delta)+\mathbb P(|X_n-X|>\delta).
$$
By the assumed <convergence in probability>, the error tends to zero. Therefore
$$
F(c-2\delta)\leq\liminf_nF_n(c_n)\leq\limsup_nF_n(c_n)\leq F(c+2\delta).
$$
The <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>.
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