8.2. Uniform Convergence
Example 8.2.2 (d): Pointwise vs Uniform Convergence
Not true. We already met the sequence fn(x) = max(n - n2 |x - 1/n|, 0) which converges pointwise to zero on the closed, bounded (i.e. compact) interval [0, 1] but not uniformly.
The sequence fn(x) = xn on [0, 1] would be another case in point.