Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f,f_n:X\to\,Y$ with $f_n\rightrightarrows\,f$ on $X$. Show that $D(f)\subset\bigcup_{n=1}^{\infty}D(f_n)$, where $D(f)$ is the set of discontinuities of $f$.

$(f_n)$ is uniformly convergent to $f$, then $(f_n)$ is pointwise convergent. from pointwise convergent, for all $\epsilon>0$ and $x\in\,X$, there exists a $\delta(\epsilon,x)$ such that $\rho(f_n(x_m),f(x))<\epsilon$ whenever $d(x_m,x)<\delta$

sorry, not pointwise convergent. it should be $f_n$ continuous at $x\in\,X$

I kinda see how it relates to discontinuities, but not sure how to connect the result with my short argument