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I am collecting some easy problems for my students and now I am facing to the following problem:

Prove that the function $$f(x)=\left(1+\frac{1}{x}\right)^x$$ is increasing in $(0,+\infty)$.

Undoubtedly, they will solve it by using the logarithmic differentiation. I am wonder what can I do if someone wants me to verify it just by doing the definition of increasing function? I think , I am missing somethings here around. Light my way. Thanks!

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Well, I tried to adapt the proof here, planetmath.org/11nnIsAnIncreasingSequence.html by substituting $n$ and $n+1$ with $\frac{n}{m}$ and $\frac{n+1}{m}$, raising everything to power $m$ and using generalized Bernoulli's inequality. So it would hold for rationals, then by some brilliant continuity argument everything would work out. The generalized Bernoulli's inequality is apparently proven using derivatives so I don't really know if this would be acceptable. –  E.Lim Dec 16 '12 at 20:05
    
@E.Lim Oh, I just see that you already used the Bernoulli inequality. And I share your concern whether it is elementary. –  WimC Dec 16 '12 at 20:29
    
Great question for committed teachers! –  amWhy Apr 16 '13 at 0:56

1 Answer 1

up vote 3 down vote accepted

You can use the Bernoulli inequality $(1+x)^\alpha \geq 1+\alpha x$ for real $x>-1$ and $\alpha \geq 1$. Then for $x>0$ and $\alpha \geq 1$

$$ \alpha x \log(1+\frac{1}{\alpha x}) = x\log(1+\frac{1}{\alpha x})^\alpha \geq x\log(1+\frac{1}{x}). $$

Edit: See E.Lim's comment above. The use of Bernoulli's inequality could be questionable.

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