Let $(f_n(x))_n$ sequence of continuous functions converging to $f:R\to R$ (continuous). Assume $|f_n-f|$ is bounded function. Then show $f_n-f$ is uniformly bounded.
Remark: It seems the uniformity should be done with respect to $n$.
The example I have in mind is $f_n(x)$ is a triangle of height $n$ over $[0,1/n]$ and $f_n(x)$ is zero elsewhere. Then $f_n(x)\to 0$, $f_n$ continuous and not uniformly bounded but wait ... then I don't understand the question.
What is annoying me is the implication line 3 of the following image :
