Sequences have come up previously in the discussion of topology, and they may have also come up back in February when I was discussing real numbers, but I'm not sure if I've ever formally defined sequences. It turns out that the definition is really simple. A sequence is a function which has a domain of the natural numbers. We could write the sequence as f(n), using n to remind us that it's just defined for natural numbers, but we usually write it as an instead.
The codomain of the sequence could potentially be anything. In the development of topology, the codomain of the sequence was often a set in some metric space. In the proof that perfect sets in Euclidean spaces are uncountable, we used two sequences of points in the Euclidean space, a sequence of neighborhoods, and a sequence of closed sets. Now that our focus is on sequences, the codomain will always be individual points in a metric space, and often in a more specific space such as the complex plane or the real number line.
The big question with a sequence is where is it going? As n gets big, does the sequence of points have some tendency? There are three basic possibilities, depending on both the sequence and the space. First, the sequence could become near a single point. Second, for a sequence of real numbers, the sequence could just increase forever. Third, the sequence could never settle down in a single direction.
If the sequence gets and stays near a single point, we can say that the sequence converges to that point. Convergence has a strict definition, which I think is beautiful, even though some people seem to hate it. If we are in some metric space, which has some distance function d(p,q) between any two points in the metric space, any convergent sequence converges to a point L, which is called the limit of the sequence. Just because L is the limit of a sequence doesn't mean that any point in the sequence actually equals L. It just means that the points in the sequence get close to L, and that they get closer as n gets bigger.
For example, consider the sequence defined by an=1/n. We are of course familiar with this as a set from topology, but the set doesn't necessarily have a particular ordering. We are now looking at it as a sequence, which means that it is a function with domain of the natural numbers, and has an order based on the order of the natural numbers. It should be clear, both from the discussion of topology and from general observation, that the limit of this sequence is 0. That is, as n increases, an gets closer to 0. No point in the sequence ever equals 0, but that's okay because the sequence gets as close as you would like to 0.
I keep saying the sequence "gets near" the limit. It's fair to ask, how near? The answer is, as near as you want to get. Pick some distance from the limit, and call that distance ε. Then we are interested in values of n such that d(an,L) is less than ε. In particular, we are interested in a minimum value N, such that for all natural numbers greater than N, d(an,L)<ε. Looking at the sequence an=1/n, if we picked ε=1/100, for example, then d(an,0)<ε for any value of n≥101. If you choose a different ε, there's a different start point, but for any positive distance ε, there is always a minimum value of N, such that for any n≥N, the distance from an to L is always less than ε.
And this is the formal definition of the limit of sequence. A sequence an in some metric space has limit L, if for any positive real number ε, there exists a natural number N, such that for any value of n greater or equal to N, the distance from an to L is less than ε. Symbolically, the last sentence is equivalent to (∀ε>0)(∃N∈ℕ)(∀n≥N) d(an,L)<ε.
A straightforward conclusion from the definition of a limit is that if a sequence has a limit, it must be unique. This can be proved by assuming that the sequence has two limits, and then using ε and the triangle theorem. If a sequence an has limits L and M, then d(L,M)≤d(an,L)+d(an,M)<2ε. Since ε is arbitrarily small, the distance between L and M is arbitrarily small, which means they must be equal, because d(L,M)=0 only if L=M. You can also argue that if L is the limit of the sequence, no other point can be the limit, because the sequence always gets within distance ε of L, and since any other point is a fixed distance from L, you can always choose ε less than the distance to that point.