Dini's theorem
Normally pointwise convergence is too weak to be uniform — but if you stack the deck with enough good structure, pointwise convergence can be promoted to uniform for free. Dini's theorem identifies a clean such setting: a monotone sequence of continuous functions, converging pointwise to a continuous limit, on a compact set. Under those four conditions, the convergence is automatically uniform.
Theorem: let K be compact, let f_n be continuous on K, suppose f_n -> f pointwise with f continuous, and suppose the sequence is monotone in n at each point (either f_n(x) increasing up to f(x) for every x, or decreasing down to it). Then f_n -> f uniformly on K. Equivalently, the 'error' functions g_n = |f_n - f| decrease monotonically to 0 pointwise, and on a compact set a decreasing sequence of continuous functions tending pointwise to 0 must tend to 0 uniformly.
Every hypothesis is genuinely needed. Drop compactness and x^n on [0,1) decreases to 0 pointwise but not uniformly. Drop continuity of the limit and x^n on [0,1] is monotone with a discontinuous limit, again not uniform. Drop monotonicity and the moving-bump examples fail. The proof uses an open-cover argument: for each epsilon the sets where g_n < epsilon eventually cover K, and compactness extracts a finite stage that works everywhere.
On [0, 1] let f_n(x) = x^n. This decreases monotonically (x^{n+1} <= x^n) to the limit f, which is 0 on [0,1) and 1 at x = 1. The limit is NOT continuous, so Dini's theorem does not apply, and indeed convergence is not uniform. Replace [0,1] by [0, 1/2]: now the limit is the continuous function 0, all hypotheses hold, and x^n -> 0 uniformly there.
Dini needs a CONTINUOUS limit; x^n on [0,1] shows why.