Limits of Sequences, their Convergence and Divergence defined, in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

Limit of a Sequence:

We say that the sequence {an} converges to the real number L, or has the limit L, and we write:

provided that an can be as close to L as we please merely by choosing n to be sufficiently large. That is, given any number

there exists an interger N such that

Here we see that when the n terms of an infinite sequence go to infinity and they equal a real number L, then the series converges to that limit value. And also, if we subtract the L value from the nth term in the sequence, and if that value is smaller than any given finite value, then it is at the limit value L.

We might geometrically represent the definition of some sequence's limit:

Here we see that as the n values increase to infinity, they converge to the limit value L

If the sequence {an} does not converge, then {an} diverges.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002,p684a.