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## Homework Statement

Let the sequence of functions fn : [0, 1] --> R be defined with fn(x) = x^n. Show that the sequence (fn(x)) converges for each x from [0, 1], but that it doesn't converge uniformly.

## The Attempt at a Solution

Now, let x be from [0, 1>. Indeed, fn converges f(x) = 0, since for every ε > 0 one can find a positive integer N such that |0 - x^n| = |x^n|< ε. If x equals 1, the sequence converges to f(x) = 1 trivially.

Now, from the conclusion above and the definition of uniform convergence, it follows that the sequence cannot converge uniformly, since it converges to one limit for x in [0, 1>, and to another one for x = 1.

I hope I did this right, thanks in advance for any help.