Question

Define uniform convergence of sequence of functions. Give an example

Answer 1

Define uniform convergence of sequence of functions. Give an example

Defifinition. A sequence of functions $f_n:X\to R$ converges uniformly to the function $f:X\to R$ if and only if $\lim\limits_{n\to+\infty}\sup\{|f_n(x)-f(x)|:x\in X\}=0$ or, equivalently, if and only if for any $\varepsilon>0$ there exists a sufficiently large integer N such that for all $x\in X$ and $n>N$ $|f_n(x)-f(x)|<\varepsilon$.

Example 1. X=[0, q], where 0<q<1, $f_n(x)=x^n$, $f(x)=0$. Then for all $x\in X$ we have

$|f_n(x)-f(x)|=x^n\leq q^n\to0$, hence $\lim\limits_{n\to+\infty}\sup\{|f_n(x)-f(x)|:x\in X\}=0$ and the sequence of functions $f_n(x)$ converges to zero uniformly.

Example 2. X=[0, 1), $f_n(x)=x^n$, $f(x)=0$. Then for all $x\in X$ we have $|f_n(x)-f(x)|=x^n$, $\sup\{x^n:x\in X\}=1$, hence $\lim\limits_{n\to+\infty}\sup\{|f_n(x)-f(x)|:x\in X\}=1$ and the sequence of functions $f_n(x)$ does not converge to f(x) uniformly.