Give a counter example: $f_n\cdot g_n$ uniformly convergent then $f_n$ or $g_n$ uniformly convergent.

Question

Let $f_n$ and $g_n$ be pointwise convergent sequences.

Is it true or not: if $f_n\cdot g_n$ is uniformly convergent then $f_n$ or $g_n$ is uniformly convergent.

Note: I think it's not true but I couldn't think of a counterexample.

Answer 1

Hint: the sequence

$$f_n(x) = \begin{cases} 1-nx & 0 \leq x < \frac{1}{n} \\ 0 & \frac{1}{n} < x \leq 1 \end{cases}$$ is pointwise but not uniformly convergent on $[0,1]$ to

$$f(x) = \begin{cases} 1 & x = 0 \\ 0 & 0 < x \leq 1. \end{cases}$$

In particular, the issue is at $0$. Try to make $g_n$ a modified version of this, such that $f_n g_n$ is constantly $0$.

Essentially the idea of this is that the $0$-ness of one function cancels out the non-uniformity of the other, letting the product converge uniformly without requiring either function to converge.

Answer 2

On $(0,\infty),$ let $f_n(x) = (x+1/n)^2, g_n(x) = 1/f_n(x).$ Then $f_n(x)g_n(x)\equiv 1$ for each $n,$ so $f_n\cdot g_n$ converges uniformly to $1$ on $(0,\infty).$ But neither $f_n$ nor $g_n$ converges uniformly on $(0,\infty).$ I'll leave the last bit to you for now; ask if you have questions.