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Let $(f_n)$ and $(g_n)$ be uniformly convergent $\Bbb K$-valued sequences of functions on $X$ with limit functions $f$ and $g$ respectively. Show that if $f$ or $g$ is in $B(X,\Bbb K)$ then $(f_ng_n)\to fg$ uniformly.

Here $B(X,\Bbb K)$ is the set of bounded functions from $X$ to $\Bbb K$, and $\Bbb K$ means $\Bbb R$ or $\Bbb C$. I think that the exercise is wrong because I found a counterexample.

Let $f_n(x):=x+\frac1n$ and $g_n(x):=\frac{\cos x}n$ and $X:=\Bbb R$, then observe that

$$\left(x+\frac1n\right)\overset{\text{uniformly}}{\longrightarrow} x,\quad\left(\frac{\cos x}n\right)\overset{\text{uniformly}}{\longrightarrow} 0$$

and obviously $g(x)=0$ is bounded. Then, if the assumptions of the exercise were true, we will have that

$$\left(\frac{\cos x}n\left(x+\frac1n\right)\right)\overset{\text{uniformly}}{\longrightarrow} 0$$

but $$\left\|\frac{\cos x}n\left(x+\frac1n\right)\right\|_\infty=\infty$$

hence $(f_n g_n)$ cannot converges uniformly in $\Bbb R$. Im right or there is something that I didnt see? It is the exercise wrong?

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    You're right, the exercise is wrong. A simpler example would be $f_n = f$ for all $n$, where $f$ is unbounded, and $g_n = \frac{1}{n}$. One needs both functions bounded, or special information about the involved functions. – 2017-01-26

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