# Homework Help: Uniform convergence

1. Jan 29, 2012

### estro

1. The problem statement, all variables and given/known data
f(x) is defined within [a,b].
$f_n(x)=\frac{\big\lfloor nf(x) \big\rfloor}{n}$

Check if $f_n(x)$ is uniform convergent.

3. The attempt at a solution
This one seems to be easy however since I didn't touch calculus for quite a time I'm not confident with my solution.

$|\frac {nf(x)-1} {n}| \leq |\frac {\big\lfloor nf(x) \big\rfloor }{n}| \leq |\frac {nf(x)+1} {n}|$ so by the squeeze theorem: $\lim_{n \rightarrow \infty}f_n(x)=f(x)$.

Now, let $x_0 \in [a,b]$ if $f(x_0) \geq 0$ then: $|f_n(x)-f(x)|=|\frac {\big\lfloor nf(x) \big\rfloor } {n}-f(x)| \leq |\frac {nf(x)+1} {n}|=\frac {1}{n} \rightarrow 0$, similarly we can show that the same equation hold when f(x_0)<0 what in turns means that we can choose N that is not dependent on x and satisfies $|f_n(x)-f(x)|< \epsilon$

Looks good?

Last edited: Jan 29, 2012