Construct sequences with uniform convergence whose product doesn't

[boombaby]

Homework Statement

Find sequences {f_n} {g_n} which converge uniformly on some set E, but such that {f_n*g_n} does not converge uniformly on E.

Homework Equations

The Attempt at a Solution

I looked at some sequences of functions known to be convergent but not uniformly convergent and tried to find {f_n} and {g_n} from that. However, I have not enough sequences at hand, I could not find a proper sequence.
I guess it is not the right way to solve this question. But I've no idea how to construct such. Any hint? Thanks

 

[Dick]

If you are looking at sequences that are not uniformly convergent, you are looking in the wrong place. Hint: is f_n(x)=x+1/n is uniformly convergent on R?

 

[boombaby]

Thanks!
your f_n is uniformly convergent on R, with limit function f(x)=x.
and f_n(x)*f_n(x) = g_n(x) = x^2+2x/n+1/(n^2) converges to h(x)=x^2, but not uniformly, since |g(n)-h(n)|>=2.
Well, I do not understand how to get this function from nowhere. However, the behavior of this function is so simple that it could be memorized easily...

 

[Dick]

To get it from nowhere, just think 'big number'*'small epsilon' isn't necessarily small if 'big number' can go to infinity.