Unbounded Monotone Sequence Diverges to Infinity/Decreasing

Theorem

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

Let $\sequence {x_n}$ be decreasing and unbounded below.

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 decreasing:

$\forall n \ge N: x_n \le x_N < H$

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 $\S 4.29 \ (5)$