# Example of a sequence $(f_n)_n$, such that $(f_ n')_n$ converges uniformly, but $(f_n)_n$ doesn't converge in any point.

I've seen the following theorem:

If $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n(c))_n$ converges to $f(c)$ for some $c\in [a,b]$, and $(f_n')_n$ converges uniformly to $g$. Then $(f_n)_n$ converges uniformly to f such that $f'=g$.

The question I have is if the convergence of $f$ for some $c$ is really necessary? I can't think of an example in which $(f_n)_n$ is a sequence of $C^1$ functions in $[a,b]$, such that $(f_n')_n$ converges uniformly to $g$, but $(f_n(c))_n$ is not convergent for all $c\in [a,b]$

Any ideas?

• $f_n(x) = x+n$. – Daniel Schepler Sep 22 '17 at 18:03
• How about $f_n = n$ – Coolwater Sep 22 '17 at 18:04
• This came up just minutes ago. – mechanodroid Sep 22 '17 at 18:08
• $f_n = (-1)^n$. – PhoemueX Sep 22 '17 at 19:44