## Definition 8.1.5: Pointwise Convergence

A sequence of functions

*{ f*with domain_{n}(x) }*converges pointwise if for each fixed***D***x*in the domain the numeric sequence_{0}**D***{ f*converges. In other words: for each fixed_{n}(x_{0}) }*x*and any given_{0}*> 0*there exists a positive integer*N*such thatfor some limit| f_{n}(x_{0}) - L | < whenever n N

*L*. Note that the limit*L*depends on*x*, while the integer_{0}*N*depends on*x*and ._{0}