Given that , (fn) be a sequence of function.
Where fn:[0,1]→R defined by fn(x)=xn .
Clearly, if x=1 then the sequence (fn(1)) converges to 1 .
If 0≤x<1 then the sequence (fn(x)) converges to 0 as we known that limn→∞xn=0 if 0≤x<1 .
Let f(x)={10if x=1if 0≤x<1
Thus the sequence of function (fn(x)) convergent to f on the set [0,1] .
If nk=k and xk=(21)k1 then
∣fnk(xk)−f(xk)∣=∣21−0∣=21 .
Therefore the sequence (fk) doesn't converge uniformly on [0,1] to f .
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