Question

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and bounded, show that $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ both exist.

   If $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and bounded, show that $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ both exist.

Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 44

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Step 1

Since $f$ is increasing, for any two real numbers $x$ and $y$ with $x < y$, we have $f(x) \leq f(y)$. This property will be key in establishing the limits at infinity. Show more…

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If $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and bounded, show that $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ both exist.

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