$f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

Question

Let $f_n(x) = n + x/n , x \in \mathbb R$. Then the seq $f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

We see that $f_n ^{'} = 1/n$ which is trivially uniformly convergent.But how to prove that $f_n$ is not uniformly convergent.

Answer 1

If $f_n$ is uniformly convergent then it is also u.c. on compact subsets (closed intervals). On a compact subset you easily prove the estimate $ |f_n(x)-f_m(x)|\ge c_0 |n-m| $ for sufficiently large m, n; where $c_0$ is a constant. Now you get easily a contradiction because $(n)$ is not a Cauchy sequence. You should have no difficulty in working out the details. Good luck!