Def/Limit (sequence)

Definition of Limit (sequence): Suppose that $(X,d)$ is a metric space, such as $\RR$ or $\CC$ with their standard metrics, for example. Suppose that $(x_i)_{i \geq 0}$ is a sequence of elements of $X$, indexed by natural numbers.

Suppose that $x \in X$. We say that $x$ is a limit of the sequence $(x_i)$, if the following condition holds:

• For all positive real numbers $\epsilon$, there exists a natural number $N$, such that $i \geq N \Rightarrow d(x, x_i) < \epsilon$.

Specifically, for sequences of real or complex numbers, the condition translates to:

• For all positive real numbers $\epsilon$, there exists a natural number $N$, such that $i \geq N \Rightarrow \vert x - x_i \vert < \epsilon$.

More generally, if $(X,T)$ is a topological space, and $(x_i)$ is a sequence in $X$, we say that $x$ is a limit of the sequence $x_i$, if the following condition holds:

• For all open subsets $U \subset X$, such that $x \in U$, there exists a natural number $N$, such that $i \geq N \Rightarrow x_i \in U$.

If $x$ is a limit of a sequence $(x_i)$, then we say or write the following:

• $x = lim_{i \rightarrow \infty} x_i$.
• "$x$ is the limit, as $i$ approaches infinity, of the sequence $(x_i)$"
• The sequence $(x_i)$ converges to $x$.

If a sequence $(x_i)$ has a limit, then we say that it is a convergent sequence.