Trying to Prove Uniform Convergence: Analysis II

[Abraham]

Homework Statement

I have a solution to the following problem. I feel it is somewhat questionable though

If f_n converges uniformly to f, i.e. f_n$\rightarrow$f as n$\rightarrow$∞ and
g_n converges uniformly to g, i.e. g_n$\rightarrow$f as n$\rightarrow$∞ ,

Prove that f_ng_n $\rightarrow$ fg

The domain is ℝ for all functions.

The Attempt at a Solution

If f and g are uniformly continuous, then for any ε>0, there is an N for which n$\geq$N implies | f_n - f | < ε. The same for g.

This is my idea:

For any given ε >0, find N such that for n>N, |f_n-f| < ε₁, and |g_n-f| < ε₁. This ε₁ should be made very small, such that:

|f_ng_n - fg| = | (f+ε₁)(g+ε₁) - fg|
= | f*g + f*ε₁ + g*ε₁ + (ε₁)² - fg|
The ε₁ terms can be made very small as needed, such that:
= | f*g + [STRIKE]f*ε₁[/STRIKE] +[STRIKE] g*ε₁[/STRIKE] + [STRIKE](ε₁)²[/STRIKE] - fg| < ε

Agree, or disagree? Secondly, is there a shorter, simpler method? Thanks.
--Abraham

 

[LCKurtz]

Homework Statement

I have a solution to the following problem. I feel it is somewhat questionable though

If f is uniformly continuous, i.e. f_n$\rightarrow$f as n$\rightarrow$∞ and
g is uniformly continuous, i.e. g_n$\rightarrow$f as n$\rightarrow$∞ ,

Prove that f_ng_n $\rightarrow$ fg

The Attempt at a Solution

If f and g are uniformly continuous, then for any ε>0, there is an N for which n$\geq$N implies | f_n - f | < ε. The same for g.

This is my idea:

For any given ε >0, find N such that for n>N, |f_n-f| < ε₁, and |g_n-f| < ε₁.

I probably should refrain from answering this because I have to leave momentarily. But my suggestion is that you need to state the question more carefully to begin with. I think you are confusing the ideas of uniform continuity and uniform convergence. You haven't told us what $f_n\rightarrow f$ means. Pointwise convergence? Uniform convergence? Any information about the domain? Problems like this are frequently addressed by adding and subtracting terms:$$|f_ng_n-fg|
=|f_ng_n-f_ng+f_ng-fg|$$ and working with that.

 

[Abraham]

Sorry, I wrote the problem incorrectly. I meant to write:

f_n converges uniformly to f, i.e. f_n→f.

I don't know why I wrote "uniformly continuous" instead...

I see what you mean though, adding and subtracting quantities. I'll start with that.