# Pointwise vs uniform convergence

1. Nov 23, 2008

### sephiseraph

Howdy Ho, partner.

I have a series of functions $${f_{n}}$$ with $$f_{n}(x) := x^{n} / (1 + x^{n})$$ and I am investigating the pointwise limit of the sequence $$f_{n}$$ over [0, 1] to see if it converges uniformly.

I found the pointwise limit f(x) to be $$f(x) = lim_{n\rightarrow\infty} x^{n} / (1 + x^{n})$$, f(x) = 0 for x in [0, 1), f(x) = 1/2 for x = 1.

My problem here comes in finding out if the sequence converges uniformly or not. Intuition tells me that, since f(x) is not continuous, there cannot be uniform convergence on [0, 1]. However we also have

$$sup\left\{|f_{n}(x) - f(x)| : x \in \left[0, 1\right]\right\} = sup\left\{x^{n} / (1 + x^{n}) : x \in \left[0, 1\right)\right\}$$ which tends to 0 as n tends to infinity, which is a sufficient condition for uniform convergence. Where have I gone wrong here or how do I make sense of this?

2. Nov 23, 2008

### morphism

Are you sure $\sup\{|f_n(x) - f(x)| \colon x \in [0,1]\}$ tends to 0?