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?