Limit of Constant Function/Limit at 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 $+\infty$.

Proof
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 $x \ge M$.

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


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