Limit of Monotone Real Function/Decreasing

Theorem
Let $f$ be a real function which is decreasing and bounded below on the open interval $\openint a b$.

Let the infimum of $f$ on $\openint a b$ be $l$.

Then:
 * $\displaystyle \lim_{x \mathop \to b^-} \map f x = l$

where $\displaystyle \lim_{x \mathop \to b^-} \map f x$ is the limit of $f$ from the left at $b$.

Proof
Let $\epsilon > 0$.

We have to find a value of $\delta > 0$ such that:
 * $\forall x: b - \delta < x < b: \size {\map f x - l} < \epsilon$

That is:
 * $l - \epsilon < \map f x < l + \epsilon$

As $l$ is a lower bound for $f$ on $\openint a b$:
 * $l - \epsilon < \map f x$

automatically happens

Because $l + \epsilon$ is not a lower bound for $f$ on $\openint a b$:
 * $\exists y \in \openint a b: \map f y < l + \epsilon$

But $f$ decreases on $\openint a b$.

So:
 * $\forall x: y < x < b: \map f x \le \map f y < l + \epsilon$

We choose $\delta = y - a$ and hence the result.