9:45 PM

We've defined convergence of a sequence. A sequence (in any metric space) converges if there is a point in that space, called the limit of the sequence, such that the points in the sequence stay arbitrarily close to the limit beyond a certain step in the sequence. If you tell me how close you want to be, I can always tell you the minimum step in the sequence to guarantee that you will be that close.

We can basically run this idea backwards, and say that for any point which is not the limit of the sequence, there exists a minimum distance from that point such that there are always some future points which are at least that far away. Okay, that was too many words. Let me try symbols. a_n is some sequence with limit L. Then for any distance ε, there exists some start point N, so for any n≥N, d(a_n,L)<ε. You can choose any positive real number for ε, and then you can always find the minimum value of N so the distance inequality holds. Now, suppose p is a point other than L. Then there always exists some ε, such that for any start point N, some n≥N exists such that d(a_n,p)>ε.

For L, every ε has one fixed N such that every n≥N works. For p, in contrast, there is some fixed ε so every N has at least one n≥N which does not work. For L, you are free to choose ε and n, but N is fixed. For p, you are free to choose N, but ε and n are fixed. I think the reversals are interesting on their own merits, but they're also important to keep in mind if you are setting out to prove that a particular point either is or is not the limit of a sequence.

So what happens if every point behaves like p? For a given sequence, assume that you can show that for any possible point in the space, the sequence spends some time away from that point. In general, these are not useful sequences. Sequences are useful because we can say what they do in the long run. What we like to say is that they converge to a limit. If no point in the space is the limit of the sequence, all we can really say is that the sequence doesn't converge.

However, if the sequence is a sequence of real numbers, we can sometimes say a little more. If the terms of the sequence tend to increase, then the sequence may not converge, but we can still speak of it having a limit. One way of expressing "tends to increase" concretely is to say that for any real number, the sequence eventually gets larger than and stays larger than that number. Pick any real number P. If based on that choice, we can find an N such that for all n≥N, a_n>P, then we can say that the sequence does not converge, but it has limit +∞. Draw a number line. Start labeling points a₁, a₂, a₃, and so on. The sequence does not have to strictly increase, so a₂ can be less than a₁, but it has to increase in general. If you choose any point on the number line and call it P, then the sequence must eventually stay on the right side of P. If you choose a bigger value for P, you would expect the sequence to cross to the right of P later, but for any possible value of P, the sequence must eventually cross to the right side and never cross back. In this case, we can say that the limit of the sequence is positive infinity.

Just to clarify, +∞ is not a real number, and the sequence does not converge. However, we can still speak of the sequence having a limit of +∞. If a sequence converges, that means that it eventually gets close to a specific point. If it has limit +∞, that means it eventually gets really big. How big? As big as you want. When will it get there? All we promise is that it will get there eventually, but that after that, it will just keep growing.

Sequences of real numbers can also have a limit of −∞, meaning the same thing in reverse. The sequence travels left on the number line as the terms increase. Infinite limits apply only to sequences of real numbers. Sequences of complex numbers, for example, can get big, but they can get big in lots of different ways, so it's hard to speak of it having a limit. However, if a_n is a sequence of complex numbers, then |a_n| is a sequence of real numbers, and it may be useful to speak of |a_n| having an infinite limit, even if a_n does not have a limit.

1 comments