Prove that the sequence $(x^n)/(1+x^n)$ is an example of a sequence of continuous functions that converges non-uniformly to a continuous limit.

The sequence again that I am trying to prove is


I know that the sequence does not converge uniformly on [0,2] because it isn't continuous. But now I don't know how to set up the second part of the question.


On $[0,1)$, the sequence converges pointwise to the continuous function $x \mapsto 0$. But not uniformly because:

$$\lim_{n \to \infty} \lim_{x \to 1} \frac{x^n}{1 + x^n} = \frac12 \neq \lim_{x \to 1} \lim_{n \to \infty} \frac{x^n}{1 + x^n} = 0$$

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