Pointwise and uniform convergence proof?

**cassius** · January 16th 2012, 02:47 AM

we have two sequences of functions on an interval I in the real line.
individually, these functions converge uniformly on I.

how do i prove that the product of these functions converges pointwise on I?

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i know the definitions of each form of convergence.
also know that if we let fn be a series of continuous functions that uniformly converges to a function f. Then f is continuous.

**FernandoRevilla** · January 16th 2012, 03:03 AM

Originally Posted by cassius

we have two sequences of functions on an interval I in the real line. individually, these functions converge uniformly on I. how do i prove that the product of these functions converges pointwise on I?

The uniform convergence is irrelevant. Take into account that

$\lim_{n\to \infty}f_n(x)=f(x)\;\wedge \;\lim_{n\to \infty}g_n(x)=g(x)\quad \forall x\in I\Rightarrow$

$\lim_{n\to \infty}f_n(x)g_n(x)=\lim_{n\to \infty}f_n(x)\cdot \lim_{n\to \infty}g_n(x)=f(x)g(x)\quad \forall x\in I$

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