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Is this true?

$f_{n} \to f$ uniformly $\Rightarrow f_{n}^{'} \to f^{'}$

No — it is false.

The counterexample

$f_{n} (x) = sin ($ nx)/n

Converges uniformly to 0, derivatives cos(nx) do not converge.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Watch it happen

Watch uniform convergence fail: xⁿ on [0, 1]

uniform convergence

Others that fail the same way

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

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