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$f$ is differential in $(0,\infty)$, $\lim_{x\to \infty}f(x) = 0$.

I need to prove:

$f'$ is strictly growing in $(0,\infty) \implies f'(x)<0$, for every $x\in(0,\infty)$.

I tried using the Fundamental theorem of calculus on a closed interval in $(0,\infty)$ but could not find how to link it to the fact that $\lim_{x\to \infty}f(x) = 0$.

How can I prove that?

Edit: Thanks to Jhon if I assume by contradiction, I get that $f'>0$ starting in some $x_0 \in (0,\infty)$ and that f is strictly growing starting at some point $y \in (x_0, \infty)$. How can I get to a contradiction using this information?

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    Isn't $f=0$ a counterexample? $0' = 0$ is growing, its limit at $\infty$ is zero, but the derivative is nowhere negative. – 2017-01-21
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    Edited the question, Thanks – 2017-01-22

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Suppose by contradiction that there exists an $x_0\in (0,+\infty)$ such that $f'(x_0)\ge0$. Since $f'$ is monotone growing in $(0,+\infty)$, you must have $$ f'(y)> f'(x_0)\ge 0 $$ for all $y> x_0$. This means that $f$ is monotone growing in $[x_0,+\infty)$. On the other hand, $f'$ is also monotone growing by hypothesis. Hence $f'$ is monotone growing and strictly positive in $[y,+\infty]$. This implies that $f$ is not only strictly growing, but growing faster and faster, which contradicts $\lim_{x\rightarrow+\infty}f(x)=0.$

Note: the statement is true only if by "monotone growing" you mean "strictly growing". Otherwise the function $f=0$ is a counterexample as pointed out by martini

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    But $f=0$ has growing derivative and limit $0$. – 2017-01-21
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    @martini Perhaps by growing, the OP means strictly growing. – 2017-01-21
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    yes you are right. I guess by "monotone growing" he means "strictly growing" – 2017-01-21
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    Could you explain why does the fact that $f$ is strictly growing in $[x_0,\infty)$ contradicts $\lim_{x \to \infty}f(x)=0$? – 2017-01-22
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    You know that $f'(x_0)\ge0$ and that $f'$ is strictly growing. Thus $f'(x)\ge 0$ for all $x>x_0$, which implies that $f$ is not only growing, but growing faster and faster, hence $\lim_{x\rightarrow\infty}f(x)$ must be $+\infty$. I agree that it is not very clear from my answer. I'll edit it to make it more readable. – 2017-01-23
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    @S.Peter are you satisfied with the answer now? – 2017-01-26