Does $\lim_{n \to \infty} f_n(x_n) = 0, X_n \overset{p}{\to} c$ implies $f_n(X_n) \overset{p}{\to} 0$?

Question

I'm trying to solve the following,

Does$$\lim_{n \to \infty} f_n(x_n) = 0, \forall x_n, X_n \overset{p}{\to} c$$ implies $f_n(X_n) \overset{p}{\to} 0$? (where $f_n$ : continuous)

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What I tried

By continuous mapping theorem, we have $f_m(X_n) \overset{p}{\to} f_m(c), \forall m$.

And since by letting $x_n=c$, I have $f_m(c) \to 0$. But can I derive from there the fact that $f_n(X_n) \overset{p}{\to} 0$?