Prove that $$1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}>\ln n$$
The base case is trivial, but I cannot show it holds for $n+1$ if it does for $n$ :/
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Matti P.
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user10000024
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2Hint: The Mean Value Theorem shows that $1/(n+1)\le\log(n+1)-\log(n)\le1/n$. – David C. Ullrich Nov 11 '21 at 14:01
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Hint : Compare the areas of some specific staircase function and the $\frac{1}{x}$-function from $1$ to $n$. – Peter Nov 11 '21 at 14:02
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I know that proof using MVT but this is supposed to be solved without "advanced" results, just with induction and basic calculus – user10000024 Nov 11 '21 at 14:04
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1What calculus are you allowed to use? (I can't think of anything more "basic" than MVT; in particular, for example, without MVT you have no Fundamental Theorem...) – David C. Ullrich Nov 11 '21 at 14:07
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No, it has to be done without integrals,derivatives,etc. Also, that question is different. – user10000024 Nov 11 '21 at 14:08
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Well, real numbers axioms, basically its meant to be solved with induction. – user10000024 Nov 11 '21 at 14:09
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3If you're not allowed to use calculus at all, then what is your definition of $\ \ln\ ?$ – Adam Rubinson Nov 11 '21 at 14:28