epsilon-N definition
This is the definition that turns the vague phrase 'the terms approach L' into something a skeptic cannot wriggle out of. Imagine a challenge game: an adversary names a tolerance epsilon, however small, and you must guarantee that from some point on, every term lies within epsilon of L. If you can always meet the challenge, the limit is L.
Precisely: a_n -> L means that for every epsilon > 0 there exists a natural number N such that for all n > N we have |a_n - L| < epsilon. Unpacking the symbols: epsilon is the demanded closeness, N is the index from which the guarantee kicks in, and |a_n - L| < epsilon says term a_n sits in the open interval (L - epsilon, L + epsilon). The N is allowed to depend on epsilon — smaller tolerances generally require waiting longer.
The order of the quantifiers is everything. It reads 'for all epsilon, there exists N, for all n > N'; swapping 'for all epsilon' and 'there exists N' would falsely demand a single N working for every tolerance at once. Mastering this definition — and its negation, used to prove divergence — is the gateway skill of rigorous analysis, because every later limit notion is modeled on it.
To prove 1/n -> 0: given epsilon > 0, choose N to be any integer with N > 1/epsilon. Then for n > N we have |1/n - 0| = 1/n < 1/N < epsilon. The challenge is met for every epsilon, so the limit is 0.
A textbook epsilon-N proof: name N explicitly in terms of epsilon, then verify the inequality.
The strict inequalities can be relaxed harmlessly: 'for all n >= N' or '|a_n - L| <= epsilon' define exactly the same limits. What cannot be relaxed is the quantifier order.