Example of convergence of pmf not implying convergence in distribution

Question

Let $X_n$ and $X$ be discrete random variables with probability mass functions $f_n$ and $f$, respectively. Is there an example where $f_n\to f$ pointwise but $X_n$ does not converge to $X$ in distribution? I read that there is no such example for continuous random variables.

Answer 1

Let all $X_n$ be identical and let $X$ be independent of $X_n$. Let $X_n$ and $X$ have the same distribution. However $X_n$ does not approach $X$, even though the distributions are identical.

It doesn't matter if the variables are discrete or continuous.