Limit at Infinity of Real Identity Function

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


$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map {I_\R} x = +\infty$
$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map {I_\R} x = -\infty$


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.