Unbounded Monotone Sequence Diverges to Infinity
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Theorem
Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be monotone, that is either increasing or decreasing.
Increasing
Let $\sequence {x_n}$ be increasing and unbounded above.
Then $x_n \to +\infty$ as $n \to \infty$.
Decreasing
Let $\sequence {x_n}$ be decreasing and unbounded below.
Then $x_n \to -\infty$ as $n \to \infty$.
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (5)$