Unbounded Monotone Sequence Diverges to Infinity/Increasing

Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ be increasing and unbounded above.


Then $x_n \to +\infty$ as $n \to \infty$.


Proof

Let $H > 0$.

As $\sequence {x_n}$ is unbounded above:

$\exists N: x_N > H$

As $\sequence {x_n}$ is increasing:

$\forall n \ge N: x_n \ge x_N > H$

It follows from the definition of divergence to $+\infty$ that $x_n \to +\infty$ as $n \to \infty$.

$\blacksquare$


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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