Uniform convergence

Abstract

In the previous two chapters we have seen what it means for a sequence $\left( {x^{(n)} } \right)_{n = 1}^{\infty }$ of points in a metric space $\left( {X,d_{X} } \right)$ to converge to a limit x; it means that $\lim_{n \to \infty } d_{X} \left( {x^{(n)} ,x} \right) < \varepsilon$ or equivalently that for every $\varepsilon > 0$ there exists an N > 0 such that $d_{X} \left( {x^{(n)} ,x} \right) < \varepsilon$ for all n > N. (We have also generalized the notion of convergence to topological spaces $\left( {X, \fancyscript {F}} \right)$ but in this chapter we will focus on metric spaces.)