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

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

$\displaystyle \exists$$\displaystyle x_0$$\displaystyle \in$$\displaystyle [a,b]$$\displaystyle \backepsilon$ as $\displaystyle x$$\displaystyle \to$$\displaystyle x_0$ that $\displaystyle \sum_{n=0}^{\infty}$$\displaystyle u_n(x)$$\displaystyle \to$$\displaystyle f$ but we have that $\displaystyle \sum_{n=0}^{\infty}$$\displaystyle u_n(x)$$\displaystyle \to$$\displaystyle f$ $\displaystyle \forall$$\displaystyle x$$\displaystyle \in$$\displaystyle [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 $\displaystyle (0,1)$ by the Weirstrass M-test

This implies that

$\displaystyle \blacksquare$

Does that look right?