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I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A_n generate A_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is that if two even permutations fail to generate A_n, then they must both be contained in some maximal subgroup of A_n.

I can't see why this is the case?

Any help much appreciated.

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If you have two elements of ${\rm A}_n$ which do not lie both in any maximal subgroup of ${\rm A}_n$, then they in particular do not lie both in any proper subgroup of ${\rm A}_n$. This in turn means that the only subgroup of ${\rm A}_n$ which contains your elements is ${\rm A}_n$ itself. Hence your two elements generate ${\rm A}_n$.

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