Limits of Sequences

Calculus deals with limits, derivatives, and integrals. For sequences the only limits of interest are those as $n\to\infty$. Derivatives don't really exist, though one can study finite differences (a whole theory for calculus of finite differences parallels the usual calculus-the major textbook in the subject is by George Boole of Boolean Algebra fame). The analog of improper integrals is given by series.

Recall the definition of

$$\lim_{x\to\infty} f(x) = L$$

For any $\epsilon>0$, there is an M such that if x>M then $\vert f(x)-L\vert<\epsilon$.

If we restrict the x's to be natural numbers we get the definition of a limit for a sequence:

Definition 2   $${\lim_{n\to \infty}a_n = L}$$ means that for any $\epsilon>0$ there is an M such that if n>M then $\vert a_n - L\vert < \epsilon$. If a sequence has a limit we say it is convergent.

The similarity of these two definitions together with the fact that you already have techniques for finding limits of functions of a real variable as $x\to\infty$ (used for finding horizontal asymptotes) makes the following lemma both useful and easy to prove:

Lemma 1   If a_n = f(n) for every n and $\lim_{x\to \infty}f(x) = L$ then $\lim_{n\to\infty}a_n = L$.

PROOF:

(Sketch) Given $\epsilon$ we get an M from $\lim_{x\to \infty}f(x) = L$. Use that same M for $\lim_{n\to\infty}a_n = L$.

Notice that this lemma only goes one way: having the function have L as limit is much stronger than having the sequence have L as limit. It is quite possible for the sequence to converge but to have no limit for the function. For example if

$$a_n=\sin(n\pi)$$

then $f(x)=\sin(x\pi)$ has no limit, but the sequence does since $\sin(n\pi)=0$ for every natural number n, so the sequence converges to 0.

Often, though, the most important theorem to uses is one which uses the least upper bound property to show that a limit exists even if you don't know what that limit is:

Theorem 2   Every bounded monotone sequence converges.

Here bounded means that there is a b such that -b<a_n <b for all $n\in\ensuremath{\mathbb{N} }$. Monotone means that the sequence either always increases or always decreases rather than mixing them up and sometimes increasing and sometimes decreasing. I'll give the proof for monotone invcreasing sequences.

PROOF:

Let S be the set of all values of the sequence

$$S=\{a_0 ,\ldots , a_n \ldots\}$$

We know that S is nonempty because it contains a₀. We know it is bounded because b is an upper bound. Now the least upper bound property (which distinguishes the real numbers from the rationals) says that any non-empty set of real numbers with an upper bound has a least upper bound. Let L be the least upper bound of S. We will show that L is the limit of the sequence.

Since L is an upper bound we know that for every n, $a_n \leq L$. Since $L-\epsilon < L$ we know that $L-\epsilon$ is not an upper bound since L was the least, so there must be some M with $a_M > L-\epsilon$. That is the M we are looking for. If n>M then $L-\epsilon < a_M < a_n \leq L$so $\vert a_n - L\vert < \epsilon$.

All of the tests for convergence of positive term series depend on this theorem, so it becomes increasingly important as the subject progresses.