Pointwise and uniform convergence
General · Mathematics
Study notes
Q: Show fₙ(x) = xⁿ does NOT converge uniformly on [0,1]. Pointwise limit f: 0 on [0,1), 1 at 1. sup|fₙ-f|: at x near 1, xⁿ stays ~1 while f = 0: sup ≥ 1/2 always! sup|fₙ-f| ↛ 0: NOT uniform! On [0,a], a<1: sup = aⁿ → 0: UNIFORM (away from 1!)!