# Thread: Uniform Convergence

1. ## Uniform Convergence

If I know that $\{f_n(x)\}$ converges uniformly on $(a,b)$ and converges pointwise on $x=a$ and $x=b$, how can I show that $\{f_n(x)\}$ converges uniformly on $[a,b]$?

2. Hi

We want the uniform convergence on $[a,b]$ that is to say : $\forall \varepsilon>0, \,\exists N \, | \, \forall n \geq N, \,\forall x \in [a,b], \, \| f_n(x)-f(x) \| < \varepsilon$

What we know :
• Uniform convergence on $(a,b)$ : $\forall \varepsilon>0, \,\exists N_0 \, | \, \forall n \geq N_0, \,\forall x \in (a,b), \, \| f_n(x)-f(x) \| < \varepsilon$
• Convergence on $x=a$ : $\forall \varepsilon>0, \,\exists N_1 \, | \, \forall n \geq N_1, \, \| f_n(a)-f(a) \| < \varepsilon$
• Convergence on $x=b$ : $\forall \varepsilon>0, \,\exists N_2 \, | \, \forall n \geq N_2, \, \| f_n(b)-f(b) \| < \varepsilon$
I suggest you try to build $N$ from $N_0$, $N_1$ and $N_2$.

3. Okay, that makes things a little clearer. Would it be enough to just let $N=max\{N_0, N_1, N_2\}$?

4. Yes, it would !

5. In fact we can strongly generalize this. If $f_n$ converge pointwise on $[a,b]$ and uniformly on $[a,b] \setminus \{ c_1,...,c_k\}$ where $c_i\in [a,b]$ then $f_n$ converges uniformly on $[a,b]$.

A more interesting question is whether we can have a similar result for infinitely many points removable as well. I am not sure yet.

EDIT: There is a simple generalization for infinitely many point but it is not an interesting generalization: if for any $\epsilon$ the set of $N$'s for each pointwise point is bounded then the sequence is actually uniformly convergent everywhere.