# Uniform convergence disprove

Why the sequence of functions $f_n(x)=\frac{1}{nx+1}$ is not uniform convergent in (0,1)? I've already prove the pointwise convergence but I can't justify why this sequence is no uniform convergent.

• math.stackexchange.com/questions/561737/…
– user42761
Commented Jun 6, 2017 at 18:27
• Possible duplicate of Uniform convergence of a family of functions on $(0,1)$
– user42761
Commented Jun 6, 2017 at 18:27
• Sorry, you are right.
– user42761
Commented Jun 6, 2017 at 18:31
• I don't know that I'd characterize the convergence as "punctual" - it seems rather slow, all things considered! (The English term for this is "pointwise".) Commented Jun 6, 2017 at 19:09
• @Chris Well, slow does not imply not punctual. You just need to get there on time.
– zhw.
Commented Jun 7, 2017 at 22:39

What happens when we take $x=1/n\in (0,1)$?
• For negating uniform convergence we need to find an $\epsilon>0$ (here we take $\epsilon=\frac12$) such that for all $N$ there exists an $n>N$ (here take any $n>N$) and an $x\in (0,1)$ (take $x=1/n$) such that $|f_n(x)-f(x)|\ge \epsilon$ (here $\frac{1}{1+nx}=\frac12 \ge \epsilon=\frac12$). Commented Jun 6, 2017 at 18:42
These functions are uniformly convergent on $(\epsilon, 1)$ for any $\epsilon > 0$ but not on the whole interval $(0,1)$.