, let be the function on the interval from [0,1] by . Which of the following statements are true?
1. The sequence { } converges pointwise on [0,1] to a limit function .
True
2. The sequence { } converges uniformly on [0,1] to a limit function .
False
3.
Don't know
Are you trying to use the Weierstrass M-Test? It isn't applicable here since we are not looking at a function defined by an infinite series.
You need to use the definition of uniform convergence: Given any , there is so that whenever , for all . (Now I realise that you are restricted to - sorry for the confusion earlier.) Can you find such an ? You should take the point into consideration.
This is the right idea, but still not exactly the case. The function converges pointwise to . So, it turns out for all , .
Have a look at a plot of the sequence below and give it another shot.
Yes, part two is false.
Your argument needs to be refined a little bit: you need to find a point that "floats" away from our function for any . Since all of the are continuous, there exists a point so that . Since this is the case, we have . Therefore, we can never find large enough so that for all . (i.e. Uniform continuity fails if we try to choose .)
In part three, try actually evaluating the integral by putting a convenient bound on it.