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Pointwise and Uniform Convergence

Definition: Letbe a subset ofand letbe a sequence of functions defined onWe say thatconverges pointwise onif exists for each point

This means thatis a real number that depends only on

Ifis pointwise convergent then the function defined by

for everyis called the pointwise limit of the sequence

More formally, given anyand there exists a natural numberdepending only onandsuch that

for every

Definition Letbe a subset ofand letbe a sequence of real valued functions defined onThenconverges uniformly toif given anythere exists a natural numbersuch thatfor everydepends only on so uniform convergence implies pointwise convergence., but the converse is not true.

Example:

Asfor eachso converges pointwise to zero.

Suppose now thatandthenandsois not uniformly convergent.

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