3.4. Upper and Lower Limits
Definition 3.4.1. Let \((x_n)\) be a sequence in \(\mathbb R\). If for every \(y\in \mathbb R\) there is \(n_0\in \mathbb N^{+}\) such that \(n\ge n_0\) implies \(x_n\ge y\), then we write \(x_n\to +\infty\). Similarly, if for every \(y\in \mathbb R\) there is \(n_0\in \mathbb N^{+}\) such that \(n\ge n_0\) implies \(x_n\le y\), then we write \(x_n\to -\infty\).
Definition 3.4.2. Let \((x_n)\) be a sequence in \(\mathbb R\), and define \(A\) to be the set of numbers \(y\in \overline {\mathbb R}\) such that \((x_{n_k})\to y\) for some subsequence \((x_{n_k})\). Let
\[\alpha = \sup_{\overline{\mathbb R}} A\quad \text{and} \quad \beta = \inf_{\overline{\mathbb R}} A.\]Then, the numbers \(\alpha\) and \(\beta\) is called the upper limit and lower limit of \((x_n)\), respectively. In this case, we write
\[\limsup_{n\to\infty}x_n=\alpha \quad \text{and} \quad \liminf_{n\to\infty}x_n=\beta.\]References
- Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). McGraw-Hill Education.