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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.

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

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