Range of Infinite Sequence may be Finite
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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