The universal approximation theorem
Cybenko proved that a network with one hidden layer and a sigmoidal function can approximate any continuous function on a bounded domain.
Why it matters
The question of whether networks can express it was settled; what stayed open was whether they can be trained to.
The proof is non-constructive: it guarantees weights exist but says neither how many units are needed nor how to find them. Hornik, Stinchcombe and White obtained a similar result independently the same year. The theorem is often cited to justify networks, though the practical limits, training and unit count, are precisely what it does not remove.