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If (fₙ) is a sequence of differentiable functions such that (fₙ') converges uniformly and (fₙ) converges pointwise, then (fₙ) converges uniformly to f, and lim fₙ' = f'. This is a deep result: passing limits through derivatives requires uniform convergence of derivatives, not just the original functions. It enables term-by-term differentiation of power series.
You know from uniform convergence that swapping limit operations requires work: uniform convergence lets you exchange a limit with an integral and preserves continuity, but neither of these facts is obvious from pointwise convergence alone. The question here is harder still: if fₙ → f, can you conclude that fₙ' → f'? That is, can you differentiate through a limit?
The naive hope fails dramatically. Consider fₙ(x) = sin(nx)/√n. These functions converge uniformly to 0 (the amplitudes 1/√n → 0), so the limit function is f = 0 and f' = 0. But differentiating gives fₙ'(x) = √n cos(nx), whose amplitude grows without bound — these derivatives do not converge at all. The functions become flat in the limit, but they oscillate wildly and increasingly rapidly along the way. So even uniform convergence of the functions is not enough to control the derivatives.
The correct theorem requires a different trade: you need uniform convergence of the derivatives fₙ', plus pointwise convergence of the functions at at least one point. If both hold, then two conclusions follow: the functions themselves converge uniformly to some limit f, and the derivatives converge uniformly to f'. You can exchange the limit and derivative: lim (d/dx fₙ) = d/dx (lim fₙ). The intuition is that uniform convergence of fₙ' controls how fast the functions change across all x simultaneously — preventing the oscillation disaster in the earlier example.
This theorem is the engine behind term-by-term differentiation of power series. A power series Σaₙxⁿ converges on an interval to some function f. You want to differentiate it by differentiating each term: (Σaₙxⁿ)' = Σnaₙxⁿ⁻¹. The interchange theorem — with the Weierstrass M-test establishing uniform convergence of the derivative series inside any compact subinterval — justifies this operation rigorously. The result is that a power series is infinitely differentiable inside its radius of convergence, and each derivative is computed term by term. This connects uniform convergence to the rich theory of analytic functions.