Real Analysis: Sequences

Sequences

A sequence is a function of the natural numbers. Sequences have special notation: if a sequence is given by some function $f : \mathbb{N} \rightarrow X$, we write it as $\{a_n\}$, where $a_n = f(n)$. In fact, we never use the regular function notation $f(n)$ and instead simply write the formula for the $n$th value in place of $a_n$. For example, the sequence of even natural numbers can be written neatly as $\{2n\}$, the sequence of odd natural numbers as $\{2n + 1\}$, and the sequence of square natural numbers as $\{n^2\}$. We can also list the first few elements of the sequence when the pattern is clear. For example, the even numbers we can simply write as $0, 2, 4, 6, \ldots$.

Sequences do not need to be expressible algebraically. For example, if we take $\{a_n\}$ to be the sequence of primes, then $a_0 = 2$, $a_1 = 3$, $a_2 = 5$, $a_3 = 7$, and so on. Any function of $\mathbb{N}$ is a sequence.

Sequences of Real Numbers

While the codomain of a sequence may be any set whatsoever, real analysis is of course focused on sequences of real numbers. The general definition in terms of arbitrary sets given above is still useful, though, since it shows us that sequences the essence of sequences generalizes beyond sequences of real numbers. For example, the above definition allows us to consider sequences of elements of $\mathbb{R}^n$. However, for now we will only be interested in sequences of real numbers unless stated otherwise.

Alternate Domains

Sequences are often defined as a function of $\mathbb{N}^+$, the positive natural numbers. In truth, the starting point is typically of little interest, as the study of sequences is concerned with what happens at the "tail" of the sequence as the index grows towards infinity. Don't be surprised to see the sequence of primes written instead starting with $a_1 = 2$, or the sequence $\left\{\frac{1}{n}\right\}$ implicitly starting with $a_1 = 1$.

Subsequences

Given a sequence $\{a_n\}$, we can construct a new sequence by selecting an infinite subset of elements from $\{a_n\}$. This subset we naturally call a subsequence. Formally, if $\{n_k\}$ is a subset of $\mathbb{N}$ where $n_0 < n_1 <\ldots$, then $\{a_{n_k}\} = a_{n_0}, a_{n_1}, \ldots$ is a subsequence of $\{a_n\}.$ For example if we take the sequence $\{(-1)^n\} = 1, -1, 1, -1, \ldots$ and index it by the sequence $\{2n\} = 0, 2, 4, \ldots$, we get the subsequence of $(-1)^0, (-1)^2, (-1)^4, \ldots = 1, 1, 1, \ldots.$ Likewise, if we take $\{a_n\}$ to instead be the sequence of prime numbers starting with $a_1 = 2$ and index it by $\{3n\}$, we get every third prime number: $5, 13, 23, 37 \ldots.$

Nested Interval and Nested K-Cell Theorems

An important theorem for the analysis of $\mathbb{R}$ is the nested interval theorem.

A nested sequence of intervals is a sequence of nonempty closed intervals $\{I_n\}$ of $\mathbb{R}$ with the following two properties:

• $I_{n+1} \subseteq I_n$
• For every $\varepsilon > 0,$ there exists an $N \in \mathbb{N}$ such that $\text{Diam}(I_n) < \varepsilon$ for all $n \geq N.$

The nested interval theorem states that if $\{I_n\}$ is a nested sequence of intervals, then their intersection $\bigcap\limits_{n=0}^{\infty} I_n$ is not empty and contains a single value.

Analogously, a nested sequence of k-cells is a sequence of nonempty closed k-cells $\{S_n\}$ in $\mathbb{R}^k$ with the following two properties:

• $S_{n+1} \subseteq S_n$
• For every $\varepsilon > 0,$ there exists an $N \in \mathbb{N}$ such that $\text{Diam}(S_n) < \varepsilon$ for all $n \geq N.$

The nested k-cell theorem states that if $\{S_n\}$ is a nested sequence of k-cells under the Euclidean metric, then their intersection $\bigcap\limits_{n=0}^{\infty} S_n$ is not empty and contains a single value.

Major theorems about the real numbers, such as the Bolzano-Weierstrass theorem and the Heine-Borel theorem, make use of the nested interval theorem, and their multidimensional analogues for $\mathbb{R}^n.$