Unbounded Monotone Sequence Diverges to Infinity

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 $\S 4.29 \ (5)$
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