Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

For each $n \ge 1$, let $f_n$ be a monotonic increasing real valued function on $[0, 1]$ such that the sequence of functions $\{f_n\}$ converges pointwise to the function $f \equiv 0$. Pick out the true statements from the following:

a. $f_n$ converges to $f$ uniformly.

b. If the functions $f_n$ are also non-negative, then $f_n$ must be continuous for sufficiently large $n$.

share|improve this question

closed as off-topic by Daniel Fischer, Michael Albanese, drhab, amWhy, Moron Jul 30 at 12:02

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Daniel Fischer, Michael Albanese, drhab, amWhy, Moron
If this question can be reworded to fit the rules in the help center, please edit the question.

@poton - Please read the FAQ for this site. This is your fifth question in which you've just pasted a homework question without stating what you've tried, or where you are having trouble, without even accepting or upvoting the answers given. Following the FAQ will encourage others to help you. –  nbubis Aug 8 '12 at 2:31
i think a is not true but have no idea about b.please help. –  poton Aug 8 '12 at 5:39
@poton Could you give your counterexample? Why do you think (a) is not true? In fact, (b) is not true. Consider $f_n(x):=0$ if $0\leq x<1/2$, $f_n(x):=1/n$ if $1/2<x\leq 1$. –  vesszabo Aug 8 '12 at 8:06
i feel that if f is not continuous then a may not be true.not sure in fact –  poton Aug 8 '12 at 11:12
I've noticed that you have asked 10 questions during last 4 days. I wanted to make sure that you are aware about the quotas 50 questions/30 days and 6 questions/24 hours, so that you can plan posting your questions accordingly. (If you try to post more questions, stackexchange software will not allow you to do so.) For more details see meta. –  Martin Sleziak Aug 9 '12 at 14:02

1 Answer 1

up vote 0 down vote accepted

For (a) you know that $f_n(0)\to 0 $ and $f_n(1)\to 0$. That is: For given $\varepsilon>0$ there is $n_0\in\mathbb N$ such that $n>n_0$ implies $|f_n(0)|<\varepsilon$ and there is $n_1\in\mathbb N$ such that $n>n_1$ implies $|f_n(0)|<\varepsilon$. Now verify that for all $n>\max\{n_0,n_1\}$ you have $|f_n(x)|<\varepsilon$ for all $x\in [0,1]$.

for (b) consider $f_n(x)=\begin{cases}\frac2nx &\mathrm{if\ }x>1-\frac1n\\\frac1nx&\mathrm{otherwise}\end{cases}$. (These are even strictly increasing.)

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.