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.

Suppose that $\{f_n\}$ and $\{g_n\}$ are $2$ sequences of unbounded functions on $S\subset\mathbb{R}$ for infinitely many $n\in\mathbb{N}$ such that $\{f_n\}\to f$ and $\{g_n\}\to g$ uniformly. Does $\{f_ng_n\}\to fg$ uniformly? I think it is not possible but i counldn't think of a counter example.

share|improve this question
    
how can a sequence be unbounded and convergent under the natural topology? –  akkkk Dec 11 '12 at 15:01
1  
@akkkk The functions are unbounded, the sequence is convergent. –  Arthur Dec 11 '12 at 15:08

1 Answer 1

up vote 7 down vote accepted

Here is a counter-example:

Take $f_n(x)=g_n(x)=x+{1\over n}$. Both of these sequences are unbounded and converge uniformly to the identity function on $\Bbb R$.

The product $f_n g_n(x)=x^2+{2x\over n}+{1\over n^2}$ converges pointwise to $h(x)=x^2$. But the convergence is not uniform, as $|f_ng_n(x)-h(x)| = |{2x\over n}+{1\over n^2}|$ cannot be made uniformly small over $\Bbb R$ for any fixed positive integer $n$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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