Limit at Infinity of Real Identity Function

Theorem
Let $I_\R: \R \to \R$ be the identity function on $\R$.

Then:


 * $(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map {I_\R} x = +\infty$


 * $(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map {I_\R} x = -\infty$

Proof
We have that the Derivative of Identity Function is $1$.

Hence, by Derivative of Monotone Function, $x$ is strictly increasing.

Now, by the definition of infinite limits at infinity, the first assertion is:


 * $\forall M \in \R_{>0}: \exists N \in \R_{>0}: x > N \implies \map f x > M$

For every $M$, choose $N = M$.

The second assertion is proved similarly.