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.