Uniform Convergence of {fn(x) = x^n} on [0,1]
Uniform Convergence:
- In order to determine if the sequence of functions {fn(x) = x^n} converges uniformly on the interval [0,1], we need to check if the limit of the sequence of functions exists and if the convergence is uniform.
Pointwise Convergence:
- First, let's check the pointwise convergence of the sequence of functions. For each fixed x in the interval [0,1], as n approaches infinity, the function fn(x) = x^n approaches 0 if 0 ≤ x < 1="" and="" fn(x)="1" if="" x="" />
Uniform Convergence Test:
- To test for uniform convergence, we need to find the supremum of the absolute difference between fn(x) and the limiting function, which is 0 for x in [0,1).
Calculating the Supremum:
- The supremum of |fn(x) - 0| occurs at x = 1 for all n. Therefore, the supremum is 1, which does not approach 0 as n approaches infinity.
Conclusion:
- Since the supremum of the absolute difference does not approach 0 uniformly on the interval [0,1], the sequence of functions {fn(x) = x^n} does not converge uniformly on [0,1].
Therefore, the statement "The sequence of functions {fn}, where fn(x)=x^n, x belongs to [0,1], n=1,2,..., converges uniformly on [0,1]" is False.