Informally, the limit, if it exists, of an infinite real sequence a₁, a₂, a₃,…is a number l with the property that a_n gets closer to l as n gets indefinitely large.

More precisely, the sequence a₁, a₂, a₃,…has the limit l if, given any ε‎ > 0 (however small), there is a number N (which may depend on ε‎) such that, for all n>N, a_n lies between l−ε‎ and l + ε‎. This is written a_n → l. A sequence’s limit, if it exists, is unique.

For example, the sequence $0,\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{15}{16},\dots$, has limit 1, and the sequence $-1,\frac{1}{2},-\frac{1}{3},\frac{1}{4},-\frac{1}{5},\dots$ has the limit 0; since this is the sequence whose nth term is (−1)ⁿ/n; this fact can be stated as (−1)ⁿ/n → 0.

There are, of course, real sequences that do not have a limit. These can be classified into different kinds.

1. (i) a_n tends to ∞, written a_n → ∞ if, given any K (however large), there is an integer N (which may depend on K) such that, for all n>N, a_n>K. For example, a_n → ∞ for the sequence a_n = n².

2. (ii) There is a similar definition for a_n →−∞, and an example is the sequence −4,−5,−6,…, in which a_n = −n−3.

3. (iii) The sequence does not have a limit but is bounded, such as the sequence $-\frac{1}{2},\frac{2}{3},-\frac{3}{4},\frac{4}{5},\dots$, in which a_n = (−1)ⁿ n/(n + 1).

4. (iv) The sequence is not bounded, but it is not the case that a_n → ∞ or a_n →−∞. The sequence 1, 2, 1, 4, 1, 8, 1,…is an example.

   If a sequence a_n converges to l, then all subsequences of a_n also converge to l.

   More generally, a sequence a_n in a metric space M converges to a limit l if d(a_n,l) → 0 as n → ∞. Thus a complex sequence (see complex number) a_n converges to the complex number l if |a_n – l|→ 0 as n → ∞. This is equivalent to Re(a_n) → Re(l) and Im(a_n) → Im(l).

   Sequential convergence determines the topology of a metric space, in the sense that a point x is in the closure of a set A if there exists a sequence a_n in A which converges to x. This is not true more generally in topological spaces.

   See algebra of limits, Bolzano-Weierstrass theorem.