A function $f_n$ either uniformly converges to its pointwise convergence function or it doesn't converge uniformly at all

Question

Wondering if the title is true.
Given that a function $f_n$ converges pointwise to a function $f$, then does this mean that if $f_n$ were to converge uniformly to some function $g$, then $f$ must equal to $g$?

Answer 1

Hint: Suppose $f_n$ converges uniformly to $f$. By definition, $\Vert f_n - f\Vert_\infty \rightarrow 0$. Now, convince yourself that this implies that for each $x$, $|f_n(x)-f(x)|\rightarrow 0$. Now, use the uniqueness of limits to arrive at your desired result.