Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = -\infty\). Give an example of a sequence that satisfies this definition.

Definitions of sequences and convergence: here, Definitions of subsequences: here.


Solution

We say that

$$ \begin{align*} \lim\limits_{n \rightarrow \infty} x_n = -\infty \end{align*} $$

If and only if for every real number \(M\), there exists an \(N \in \mathbf{N}\) such that for all \(n \geq N\), we have

$$ \begin{align*} x_n < M \end{align*} $$

An example of a sequence that satisfies this definition is

$$ \begin{align*} x_n &= -n, \\ \{x_n\} &= \{-1,-2,-3,\cdots\} \end{align*} $$

For any real number \(M\), we can choose \(N > -M\) so that for all \(n \geq N\)

$$ \begin{align*} x_n = -n \leq -N \end{align*} $$

and we know that \(-N < M\). Then

$$ \begin{align*} -n < M \end{align*} $$

as required by the definition.


References