The Tail of a Sequence of Real Numbers

# The Tail of a Sequence of Real Numbers

We will now look at an important aspect of a sequence known as the tail of a sequence.

 Definition: Let $(a_n) = (a_1, a_2, ... )$ be a sequence of real numbers. Then for any $m \in \mathbb{N}$, the $m$-Tail of $(a_n)$ is a the subsequence $(a_{m+1}, a_{m+2}, ... ) = (a_{m+n} : n \in \mathbb{N})$.

Recall that for a sequence $(a_n)_{n=1}^{\infty}$ that converges to the real number $L$ then $\lim_{n \to \infty} a_n = L$, that is $\forall \epsilon > 0$ there exists a natural number $n \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$. For any given positive $\epsilon$ we can consider the $n$-tail of the sequence $(a_n)$ to be the subsequence of $(a_n)$ such that all terms in this tail are within an $\epsilon$-distance from our limit $L$. The diagram below illustrates this concept. The following theorem tells us that the m-tail of a sequence must also converge to the limit $L$ provided the parent sequence $(a_n)$ converges to $L$.

 Theorem 1: Let $(a_n)$ be a sequence of real numbers. Then $(a_n)$ converges to $L$ if and only if for any $m \in \mathbb{N}$ the $m$-tail of $(a_n)$, call it $(a_{n_k})$ converges to $L$.