Subsequences of Sequences of Real Numbers

Definition: Let $(a_n)$ be a sequence of real numbers, and let $n_1 < n_2 < ... < n_k < ...$ where $n_i \in \mathbb{N}$ for all $i \in \mathbb{N}$. Then the sequence $(a_{n_k}) = (a_{n_1}, a_{n_2}, ..., a_{n_k}, ...)$ is said to be a Subsequence of $(a_n)$.

For example, consider the sequence of natural numbers $(n) = (1, 2, 3, ... )$. One such subsequence of the natural numbers is the sequence of all even natural numbers $(2n) = (2, 4, 6, ...)$ where $n_1 = 2$, $n_2 = 4$, …, $n_k = 2k$ Another such subsequence of the natural numbers is the sequence of all odd natural numbers $(2n - 1) = (1, 3, 5, ...)$ where $n_1 = 1$, $n_2 = 3$, …, $n_k = 2k - 1$.

We note that in both cases above, order of the terms of these subsequences are preserved since $n_1 < n_2 < ... < n_k < ...$. For example, the sequence $(4, 2, 6, 8, ... )$ is NOT a subsequence of $(n)$ since $n_1 = 4$ and $n_2 = 2$ and clearly $n_1 \neq < n_2$.

We will now look at some important theorems regarding subsequences.

Theorem 1: Let $(a_n)$ be a convergent sequence such that $\lim_{n \to \infty} a_n = A$. Then any subsequence $(a_{n_k})$ is also convergent, and $\lim_{k \to \infty} a_{n_k} = A$.

• Proof: Let $(a_n)$ be a convergent sequence such that $\lim_{n \to \infty} a_n = A$. We want to show that $\lim_{k \to \infty} a_{n_k} = A$.

• Since $(a_n)$ is a convergent sequence, then $\forall \epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - A \mid < \epsilon$. We note that since $n_1 < n_2 < ... < n_k < ...$ is an increasing sequence of natural numbers, that also $n_k ≥ k$. If we choose $k ≥ N$ then we have that $n_k ≥ k ≥ N$ and so $\mid a_{n_k} - A \mid < \epsilon$ and so $\lim_{k \to \infty} a_{n_k} = A$. $\blacksquare$

Theorem 1 above is easier to understand visually. Consider the following sequence $(a_n)$ graphed below:

Notice that any subsequence of the sequence graphed above must also convergent to the same limit $A$. This is shown with the following two subsequences plotted. The subsequence on the left is all the even terms of the original sequence, while the subsequence on the right is all the odd terms of the original sequence:

With this notion in mind, we can also describe the tail of a sequence to be a special subsequence. We note that if $(a_n)$ is convergent to $A$, then any $m$-tail of $(a_n)$ is a subsequence that is also convergent to the real number $A$.