tail of a sequence

The tail of a sequence is everything past a chosen cutoff — chop off the first N terms and keep the infinitely many that remain. If you think of a sequence as a journey, the tail is the rest of the road ahead from wherever you currently stand, ignoring the path already traveled.

Formally, for each N the N-th tail of (a_n) is the sequence (a_{N+1}, a_{N+2}, a_{N+3}, ...), itself a perfectly good sequence. The key principle is that convergence and the value of the limit are tail properties: (a_n) converges to L if and only if some (equivalently every) tail converges to L. Discarding or altering finitely many initial terms cannot change the outcome.

This is why so many statements in analysis are phrased 'for all sufficiently large n' or 'eventually' — they are assertions about a tail, and a tail is all that the limit can see. The same principle underlies the epsilon-N definition (the index N marks where the relevant tail begins) and explains why convergence is robust against finite tampering at the front of a sequence.