222 Convergence
A sequence that gets closer to a fixed element that is used for finding the limit of a function at a real number.
definition [d] (Convergence) From Kreyszig Definition 1.4-1: a sequence \((x_{n})\) in a metric space \(X = (X,d)\) is said to converge if there is an \(x \in X\) such that
- \(\displaystyle\lim_{n \to \infty} d(x_{n}, x) = 0\) .
The point \(x\) is called the limit of \((x_{n})\) and we write
- \(\displaystyle\lim_{n \to \infty} x_{n} = x\)
or simply \(x_{n} \to x\). We say that \((x_{n})\) converges to \(x\) or has the limit \(x\). If \((x_{n})\) is not convergent, it is said to be divergent.
where
- \(X = (X,d)\) is a metric space.
- \((x_{n})\) is a sequence in \(X\).
- \(x \in X\) is the limit.
- \(d\) is the metric on \(X\).
222.1 Elementary Example
222.1.1 Simple
A sequence converges when its terms approach a limit in the space. On \(\mathbb{R}\), \(1/n\) converges to \(0\).
\[ X = \mathbb{R},\quad d(x,y) = |x - y| \]
\[ x_{n} = \dfrac{1}{n},\quad \lim_{n \to \infty} x_{n} = 0 \]
\[ d(x_{n},0) = \dfrac{1}{n} \to 0 \]
where
- \(0\) is the limit of \((x_{n})\).
- \(d\) is the metric on \(X\).