Limit of Constant Function/Limit at Minus Infinity

Theorem

Let $a, b \in \R$.

Define $f : \R \to \R$ by:

$\map f x = a$ for each $x \in \R$.

Then:

$\ds \lim_{x \mathop \to -\infty} \map f x = a$

where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.

We have:

$\size {\map f x - a} = 0$ for all $x \in \R$.

So for any $\epsilon > 0$ and $M \in \R$, we have:

$\size {\map f x - a} < \epsilon$ for all $x \in \R$ with $x \le M$.

So from the definition of the limit at $-\infty$, we have:

$\ds \lim_{x \mathop \to -\infty} \map f x = a$

$\blacksquare$