## Help on uniform convergence

Will someone help me with this, I PMed Perfect Hacker and he helped me but I am still a little unclear, but he isn't here and I am impatient.

My book asked to explain what uniform convergence meant in your own words. Would someone point out any mistakes in my comprehension if there is any.

Let

$\sum_{n=0}^{\infty}$ $u_n(x)$ be convergent on the interval $[a,b]$. Furthermore let $u_n(x)$ $\in$ $\mathcal{C}$[tex]

Now my understanding of uniform convergence (I really couldn't tell if this was wrong since TPH didn't adress it directly) is that not only does

$\exists$ $x_0$ $\in$ $[a,b]$ $\backepsilon$ as $x$ $\to$ $x_0$ that $\sum_{n=0}^{\infty}$ $u_n(x)$ $\to$ $f$ but we have that $\sum_{n=0}^{\infty}$ $u_n(x)$ $\to$ $f$ $\forall$ $x$ $\in$ $[a,b]$

Next it asked to prove that

My first step was saying that not only is

But that

And also I noted that

Now we see that

Therefore

is convergent

Therefore

is uniformly convergent on $(0,1)$ by the Weirstrass M-test

This implies that

$\blacksquare$

Does that look right?