I'm new to uniform converge in sequence function, so I have:
Let $f(x)$ be a continuous differentiable function in $R$.
$f_n(x)=n(f(x+\frac{1}{n})-f(x))$.
I need to find $\lim_{n\rightarrow \infty}f_n(x)$ and to prove that for any interval $[a, b], f_n(x)$ converge uniformly.
Now, is that correct that: for every $x$ in $R$, $\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}n(f(x+\frac{1}{n})-f(x))=0$ therefore the sequence function converge to zero. In addition, $f'(x) =\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{n \to\infty} \frac{f(x+\frac{1}{n}) - f(x)}{\frac{1}{n}}=\lim_{n \to \infty} f_n(x)=0$ so for every $x$, so we have a constant function and therefore converge uniformly. Can you help me with this? Thanks!!