Let G be a finite group of order n with identity element e. If $\displaystyle a_1, ..., a_n$ are n elements of G, not necessarily distinct, prove that there are integers p and q with $\displaystyle 1 \leq p \leq q \leq n$ such that $\displaystyle a_p a_{p+1} ... a_q = e$.

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I have not got very far in proving this apart from a few basic cases. If any of the elements on the list are the identity, it is trivial. This means that you can assume it is a list of length n which contains at most (n-1) distinct elements of G. So at least one element occurs twice. I took some cases after this, but every case is difficult and seems to lead to taking more cases

. I'm sure there is a simpler way... can anyone help?