(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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}[itex]\rightarrow[/itex]fas n[itex]\rightarrow[/itex]∞ and

g_{n}converges uniformly to g, i.e.g_{n}[itex]\rightarrow[/itex]fas n[itex]\rightarrow[/itex]∞ ,

Prove thatf_{n}g_{n}[itex]\rightarrow[/itex] fg

The domain is ℝ for all functions.

3. The attempt at a solution

If f and g are uniformly continuous, then for any ε>0, there is an N for which n[itex]\geq[/itex]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| < ε_{1}, and |g_{n}-f| < ε_{1}. This ε_{1}should be made very small, such that:

|f_{n}g_{n}- fg| = | (f+ε_{1})(g+ε_{1}) - fg|

= | f*g + f*ε_{1}+ g*ε_{1}+ (ε_{1})^{2}- fg|

The ε_{1}terms can be made very small as needed, such that:

= | f*g + [STRIKE]f*ε_{1}[/STRIKE] +[STRIKE] g*ε_{1}[/STRIKE] + [STRIKE](ε_{1})^{2}[/STRIKE] - fg| < ε

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

--Abraham

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# Trying to Prove Uniform Convergence: Analysis II

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