Answer to Question #170557 in Real Analysis for Prathibha Rose

Question

Let {f_n} be a sequence of functions defined on S .show.that there exists a.function f such that f_n converges to f uniformly on S if and.only if the caught condition is satisfied

Answer 1

The sequence of functions {fn} defined on S [a, b] {let} converges uniformly on S [a, b] if and only if for every ε > 0 and for all x ∈ [a, b], there exists an integer N such that  

$$|f_{n+p}(x)-f_n(x)|<\epsilon,\ \ \ \ \forall \ n\geq N$$

Proof. Let the sequence {fn} uniformly converge on [a, b] to the limit function f, so that for a given ε > 0, and for all x ∈ [a, b], there exist integers $n_1,n_2$ such that 

$$|f_{n}(x)-f(x)|<\epsilon/2,\ \ \ \ \forall \ n\geq n_1$$

and

$$|f_{n+p}(x)-f(x)|<\epsilon/2,\ \ \ \ \forall \ n\geq n_2$$

Let N= max($n_1,n_2$)

$$\Rightarrow|f_{n+p}(x)-f_n(x)|\leq|f_{n+p}(x)-f(x)|+ |f_{n}(x)-f(x)|$$

$\Rightarrow|f_{n+p}(x)-f_n(x)|< \epsilon/2+\epsilon/2=\epsilon, \ \ \ \ \forall\ n\geq N$

$$\text{Hence}\ \ \ |f_{n+p}(x)-f_n(x)|<\epsilon,\ \ \ \ \forall \ n\geq N$$

(Cauchy’s Criterion for Uniform Convergence)