Divergent Real Sequence to Negative Infinity/Examples/Minus Square Root of n

Example of Divergent Real Sequence to Negative Infinity

Let $\sequence {a_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$a_n = -\sqrt n$

Then $\sequence {a_n}$ is divergent to $-\infty$.

Proof

Let $H \in \R_{>0}$ be given.

We need to find $N \in \R$ such that:

$\forall n > N: -\sqrt n < -H$

That is:

$\forall n > N: \sqrt n > H$

That is:

$\forall n > N: n > H^2$

We take:

$N = H^2$

and the result follows.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (3) \ \text{(ii)}$