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I'm trying to solve this rather simple question, but no luck.

Let f be continuous and injective in $\mathbb R$.
prove that if $ \lim_{x\to \infty}{f(x)}=\infty $ than ${f}$ is monotonic increasing in $\mathbb R$.

My approach is to negatively assume that given some a and b such that ${a < b}$ then ${f(a) \ge f(b)}$, but this is where I got stuck. I don't know how to continue from here.

Any help would be appreciated!

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A good start. By injectivity, $f(a) > f(b)$. Now, what does the continuity tell you about $f([b,\infty))$? –  Daniel Fischer Mar 22 at 11:16

1 Answer 1

As you has said, let's suppose that there exist two real numbers $a<b$ such that $f(a)\geq f(b)$. Since $f$ is injective, $f(a)>f(b)$. Now we have $\lim_{x\rightarrow\infty}=\infty$, so there exists $c>b$ such that $f(c)>f(a)$. The continuity of $f$ grants a point $d\in (b,c)$ such that $f(d)=f(a)$, a contradiction since $f$ is injective.

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