1. ## uniform convergence

is this true or false? Explain why. I think it is not true because the interval is unbounded. However, I cannot explain why.

2. Originally Posted by xyz

is this true or false? Explain why. I think it is not true because the interval is unbounded. However, I cannot explain why.
It is true. Here is a hint

$f_n(t)=\frac{1}{t\ln(n)}$ the largest value of any sequence member is given when $t=1$ i.e $f_n(1)\gef_n(t)$ for all $t \in [1,\infty]$

3. ## uniform convergence

If thats true, I am not quite sure how this problem is done. The complete problem is

I appreciate your help a lot. The next thing, is to take limit of n on both sides. Does that mean right hand limit at 0 and left hand limit at 0? Please let me know how the problem ends up with 1 as value of integral when it should be 0?

4. It's true that uniform convergence allows you to interchange the limit and the integral, but only on a bounded interval $[a,b]$. (The proof of the theorem uses the value $b-a$, which makes no sense if $b=\infty$.)