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
- Problem Statement Source: Aleph 0