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Abraham
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Homework Statement
I have a solution to the following problem. I feel it is somewhat questionable though
If fn converges uniformly to f, i.e. fn[itex]\rightarrow[/itex]f as n[itex]\rightarrow[/itex]∞ and
gn converges uniformly to g, i.e. gn[itex]\rightarrow[/itex]f as n[itex]\rightarrow[/itex]∞ ,
Prove that fngn [itex]\rightarrow[/itex] 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[itex]\geq[/itex]N implies | fn - f | < ε. The same for g.
This is my idea:
For any given ε >0, find N such that for n>N, |fn-f| < ε1, and |gn-f| < ε1. This ε1 should be made very small, such that:
|fngn - 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
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