For n Element N let fn x xn1 xn x element 01 prove that t

For n Element N. let f_n (x) = x^n/(1 + x^n), x element [ 0,1]. prove that the sequence { f_n} does not converge uniformly on [ 0,1].

Solution

if fn is a sequence of function then {fn} converges to f uniformly iff

1. fn converges to f pointwise

2.let Mn =sup{ |fn(x)-f(x)| ;xbelongs to E}

Mn converges to zero.

Given fn(x)= x^n/(1+x^n) ,E=[0,1]

now lim fn(x)=0 clearly as lim x^n tends to 0 in [0,1]

consider Mn=sup{|x^n/(1+x^n)| ; xbelongs to[0,1]}

as x tends to1,x^n/1+x^n tends to 1/2 so

Mn>=1/2

thus Mn does not tends to 0.

fn does not converge uniformly..