Range of Infinite Sequence may be Finite

Theorem

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence.

Then it is possible for the range of $\sequence {x_n}$ to be finite.

Proof

Consider the infinite sequence $\sequence {x_n}_{n \mathop \in \N}$ defined as:

$\forall n \in \N: x_n = \dfrac {1 + \paren {-1}^n} 2$

Thus:

$\sequence {x_n}_{n \mathop \in \N} = 1, 0, 1, 0, \dotsc$

Hence the range of $\sequence {x_n}$ is $\set {0, 1}$, which is finite.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences