Show that one-sided limits always exist for a monotone function on an interval $[a,b]$.
Me attempt:
1) If a function is monotone on an interval $[a,b]$, then $f(a)\le f(x) \le f(b)$ for $x\in[a,b]$. Therefore if there exists left-hand (right-hand) limit of this function at a given point, then it must be finite. Now we must show that a both left-hand and right=hand limits exist. We use the fact that if we take a sequence $(x_n)\rightarrow c^{+} \in [a,b]$ then $f(x_n)$ is bounded and monotone and therefore it is convergent.
Best Answer
Outline: The general idea is right, but probably a lot more detail is expected. Note that there are two (very similar) cases, monotone non-decreasing and monotone non-increasing. In what follows, we deal with monotone non-decreasing.
It is useful to treat limits from the left and limits from the right separately. We look at the limit from the left. Let $c\in(a,b]$. We want to show that $\lim_{x\to c^{-}} f(x)$ exists. Let $U$ be the set of all $x$ in our interval such that $x\lt c$. Let $V$ be the set of all $f(x)$, where $x$ ranges over $U$. Show that $V$ is non-empty and bounded above. Then $V$ has a supremum $v$. Show that $\lim_{x\to c^{-}} f(x)=v$.
Alternately, one can use sequences.