I'm ask to prove whether or not the sequence of functions $\displaystyle {f_n}$ defined on [-1,1] is pointwise convergent, where $\displaystyle f_n = (1-|x|)^n$ for each natural number n. I was hoping someone could look over my proof and see if I made any mistakes. The cases where x=0 and |x|=1 are obvious, so consider those as already proven.

Proof: Let $\displaystyle \epsilon >0$ be given and let $\displaystyle x \in (-1, 1)$ where $\displaystyle x \ne 0$. Then, $\displaystyle 0<1-|x|<1$. Let N be a positive integer such that $\displaystyle N> \frac{|ln(\epsilon)|}{|ln(1-|x|)|}$. Then, $\displaystyle \epsilon >(1-|x|)^N$ (I'm not 100% sure this is right, so please correct me if I'm wrong). So, for any positive integer $\displaystyle n>N $, we have $\displaystyle |(1-|x|)^n-0|=(1-|x|)^n<(1-|x|)^N<\epsilon$. Therefore, $\displaystyle {f_n}$ converges pointwise to the zero function on (-1,1)-{0}.