Limit of Monotone Real Function

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Theorem

Increasing Function

Let $f$ be a real function which is increasing and bounded above on the open interval $\openint a b$.

Let the supremum 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 $b$.


Decreasing Function

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 $b$.