We were introduced to sequences in calculus, and we can extend the notion of the limit of a sequence to metric spaces. Sequences provide an alternate way to describe many ideas in metric space. For example, we will see that we can characterize continuity in terms of sequences, and we can use sequences to determine open and closed sets.
Recall from calculus that a sequence of real numbers is a list of numbers in a specified order. We write a sequence \(a_1\text{,}\)\(a_2\text{,}\)\(\ldots\text{,}\)\(a_n\text{,}\)\(\ldots\) as \((a_n)_{n \in \Z^+}\) or just \((a_n)\text{.}\) If we think of each \(a_n\) as the output of a function, we can give a more formal definition of a sequence as a function \(f: \Z^+ \to \R\text{,}\) where \(a_n = f(n)\) for each \(n\text{.}\)
A sequence \((a_n)\) of real numbers converges to a number \(L\) if we can make all of the numbers in the sequence as close to \(L\) as we like by choosing \(n\) to be large enough. Once again, this is an informal description that we need to make more rigorous. As we saw with continuous functions, we can make more rigorous the idea of “closeness” by introducing a symbol for a number that can be arbitrarily small. So we can say that the numbers \(a_n\) can get as close to a number \(L\) as we want if we can make \(| a_n - L | \lt \epsilon\) for any positive number \(\epsilon\text{.}\) The idea of choosing \(n\) large enough is just finding a large enough fixed integer \(N\) so that \(| a_n - L | \lt \epsilon\) whenever \(n \geq N\text{.}\) This leads to the definition.
Definition9.1.
A sequence \((a_n)\) of real numbers has a limit \(L\) if, given any \(\epsilon \gt 0\) there exists a positive integer \(N\) (depending only on \(\epsilon\)) such that
\begin{equation*}
| a_n - L | \lt \epsilon \ \text{ whenever } \ n \geq N\text{.}
\end{equation*}
When a sequence \((a_n)\) has a limit \(L\text{,}\) we write
or just \(\lim a_n = L\) (since we assume the limit for a sequence occurs as \(n\) goes to infinity) and we say that the sequence \((a_n)\) converges to \(L\text{.}\)
Example9.2.
We can draw a graph of a sequence \((a_n)\) of real numbers as the set of points \((n,a_n)\text{.}\) In this way we can visualize a sequence and its limit. By definition, \(L\) is a limit of the sequence \((a_n)\) if, given any \(\epsilon \gt 0\text{,}\) we can go far enough out in the sequence so that the numbers in the sequence all lie in the horizontal band between \(y = L -\epsilon\) and \(L+ \epsilon\) as illustrated in Figure 9.3 for the sequence \(\left(\frac{n}{1+n}\right)\text{.}\)
Figure9.3.The limit of the sequence \(\left(\frac{n}{1+n}\right)\text{.}\)
To verify that the limit of the sequence \(\left(\frac{n}{1+n}\right)\) is \(1\text{,}\) we start with \(\epsilon \gt 0\text{.}\)
Scratch work. Now we need to find \(N\) so that \(n \geq N\) implies \(\left| \frac{n}{1+n} - 1 \right| \lt \epsilon\text{.}\) Just as with our continuity example, this work is not part of the proof, but shows how we go about finding the \(N\) we need. To make \(\left| \frac{n}{1+n} - 1 \right| \lt \epsilon\) we need
So the sequence \(\left(\frac{n}{1+n}\right)\) has a limit of \(1\text{.}\)
Definition 9.1 only applies to sequences of real numbers. Ultimately, we want to phrase the definition in a way that allows us to define limits of sequences in metric spaces and topological spaces. So we have to reformulate the definition in such a way that it does not depend on distances.
Recall that \(| x-y |\) defined a metric \(d_E\) on \(\R\text{,}\) that is
So we can rephrase the definition of a limit of a sequence of real numbers as follows.
Definition9.4.Alternate Definition.
A sequence \((a_n)\) of real numbers has a limit \(L\) if, given any \(\epsilon \gt 0\) there exists a positive integer \(N\) (depending only on \(\epsilon\)) such that
Once we have described a limit of a sequence in terms of a metric, then we can extend the idea into any metric space.
Definition9.5.
A sequence in a metric space \((X,d)\) is a function \(f: \Z^+ \to X\text{.}\)
If \(f\) is a sequence in \(X\text{,}\) we write the sequence defined by \(f\) as \((f(n))\text{,}\) where \(n \in \Z^+\text{.}\) We also use the notation \((a_n)\text{,}\) when \(a_n = f(n)\text{.}\) As long as \(X\) has a metric defined on it, we can then describe the limit of a sequence.
Definition9.6.
Let \((X,d)\) be a metric space. A sequence \((a_n)\) in \(X\) has a limit \(L \in X\) if, given any \(\epsilon \gt 0\) there exists a positive integer \(N\) (depending only on \(\epsilon\)) such that
In other words, a sequence \((a_n)\) in a metric space \((X,d)\) has a limit \(L \in X\) if \(\lim d(a_n,L) = 0\) — or that the sequence \(d(a_n,L)\) of real numbers has a limit of \(0\text{.}\) Just as with sequences of real numbers, when a sequence \((a_n)\) has a limit \(L\text{,}\) we say that the sequence \((a_n)\) converges to \(L\text{,}\) or that \(L\) is a limit of the sequence \((a_n)\text{.}\)
Preview Activity9.1.
(a)
Explain why the sequence \(\left(\frac{1}{n}\right)\) converges to 0 in \(\R\) using the Euclidean metric \(d_E\text{,}\) where
