How do I prove the following statement? I'm reading a book in Real Analysis and one of the proofs given is complete if I have the following:
Let $F : \mathbb{R} \to \mathbb{R}$ be increasing and right continuous. As $F$ is increasing (monotone), $\lim_{x \to \infty} F(x)$ and $\lim_{x \to -\infty} F(x)$ are well-defined in $\overline{\mathbb{R}}$ (the extended real line).
I feel like this should be simple, for some reason I'm having difficulty with it. (One proof is: As $F$ is increasing, using the monotone convergence theorem every increasing sequence $x_n \to \infty \implies F(x_n) \to \infty$ or $F(x_n)$ converges to a member of $\mathbb{R}$ (this happens when $\lim_{x \to \infty} F(x)$ is bounded). It is easy to prove that if we have increasing sequences $x_n, y_n \to \infty$ and $F(x_n), F(y_n)$ do not converge to the same value in $\overline{\mathbb{R}}$, we have a contradiction to $F$ increasing. But this proof seems too routine, and a bit boring. Is there another exciting proof?)
Best Answer
Define $L = \sup \{ f(x) \mid x \in \Bbb R \} \in \overline{\Bbb R}$ (i.e. $L = +\infty$ if the set $\{ f(x) \mid x \in \Bbb R \}$ does not have an upper bound, otherwise $L \in \Bbb R$ is the “usual” supremum).
For any $K < L$
Conclude that $\lim_{x \to \infty} f(x) = L$.