Sequences:Tails

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 $\forall \varepsilon >0$ there exists a natural number $n\in \mathbb {N}$ such that if Failed to parse (syntax error): {\displaystyle n ≥ N} then $\mid a_{n}-L\mid <\varepsilon$. For any given positive $\varepsilon$ 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 $\varepsilon$-distance from our limit $L$. The diagram below illustrates this concept.

<img src="http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png" alt="Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png" class="image" />

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 $L$.