- #1

spenghali

- 14

- 0

## Homework Statement

1.) Prove that if { [tex]f_{n}[/tex] } is a sequence of functions defined on a set D, and if there is a sequence of numbers [tex]b_{n}[/tex], such that [tex]b_{n}[/tex] [tex]\rightarrow[/tex] 0, and | [tex]f_{n}[/tex](x) | [tex]\leq[/tex] [tex]b_{n}[/tex] for all x [tex]\in[/tex] D, then { [tex]f_{n}[/tex] } converges uniformly to 0 on D.

2.) Prove that if { [tex]f_{n}[/tex] } is a sequence of functions defined on a set D, and if { [tex]f_{n}[/tex] } converges uniformly to zero on D, then { [tex]f_{n}[/tex]([tex]x_{n}[/tex]) } converges to zero for every sequence { [tex]x_{n}[/tex] } of points of D.

## Homework Equations

Definition of Uniform Convergences:

{ [tex]f_{n}[/tex] } is said to converge uniformly on D to a function f if, for each [tex]\epsilon[/tex] > 0, there is N such that,

| f(x) - [tex]f_{n}[/tex](x) | < [tex]\epsilon[/tex] whenever x [tex]\in[/tex] D and n>N.

## The Attempt at a Solution

1.) SO for this one, it seems that i can just pick N = [tex]b_{n}[/tex] = [tex]\epsilon[/tex] and then this theorem follows immediately from the definition of uniform convergence.

2.) Similarly, for this one, because [tex]x_{n}[/tex] is just a sequence of points in D, then we just replace x with [tex]x_{n}[/tex] and the proof will also follow immediately from the definition of uniform convergence.

Am I on the right track with these? They both seem some what trivial.