Is the product of two uniformly convergent sequence of functions uniformly convergent?

Question

I am doing the exercise V.1.7 of Amann and Escher's Analysis. In the exercise I am required to prove that the product of two uniformly convergent sequence of functions is uniformly convergent if just one of them is bounded. But I found a counter example: $$f_n=1/n,\quad g_n=x+1/n,\quad x\in \mathbb R$$ They are both uniformly convergent and $f_n$ is bounded but the product doesn't. Is there a mistake in the exercise or there are more conditions needed?