The functions f_n(x) = x^n on [0, 1] converge pointwise. Give the limit function, compute sup |f_n - f| over [0, 1] for each n, and give the largest a for which the convergence on [0, a] is uniform.

The functions f_n(x) = x^n on [0, 1] converge pointwise. Give the limit function, compute sup |f_n - f| over [0, 1] for each n, and give the largest a for which the convergence on [0, a] is uniform.

Approach: Write down the pointwise limit first, then evaluate the supremum of the difference over the whole interval before restricting the domain and re-evaluating it.

The pointwise limit is f(x) = 0 for x in [0, 1) and f(1) = 1, the supremum norm sup |f_n - f| equals 1 for every n, and the convergence is uniform on [0, a] for every a < 1. On [0, 1) the difference is x^n, whose supremum approaches 1 as x approaches 1, so the supremum norm stays at 1 and uniform convergence fails on the closed interval. On [0, a] the difference is bounded by a^n, which tends to zero, so the convergence is uniform there for any a strictly below 1, and no single largest such a exists. The structural reason is that a uniform limit of continuous functions is continuous, and the limit here is discontinuous at 1, so uniform convergence on [0, 1] was impossible from the start.

Follow-up: Do the derivatives n x^{n-1} converge pointwise on [0, 1], and what does that say about differentiating a limit term by term?

Key concepts: pointwise convergence, uniform convergence, supremum norm, continuity of the limit.