$\ln(n)/n<1/2$ proof without calculus or any kind of advanced mathematics

Question

Is it possible to show that $\ln(n)/n<1/2$, for all natural numbers $n$ without using calculus, but just some elementary math? Induction is allowed. I was trying to show equivalently that $n-\ln(n^2)>0$, but without successes so far. Thank you.

Answer 1

To show $\ln(n)/n < \frac12 \iff n^2 < e^n$, lets try induction.

Suppose this true for some $k > 2$, then we have $(k+1)^2\ge e^{k+1} \implies k^2+2k+1 \ge ee^k > ek^2 \implies 2k+1 \ge (e-1)k^2> k^2 $ $\implies 1\ge k(k-2)$ which is clearly false for $k> 2$.

All that remains is to show $n^2 < e^n$ for $n=1, 2, 3$ which is easy.

Answer 2

Induction proof without calculus!

Each natural number $n$ falls between $e^k<n\leq e^{k+1}$ for some integer $k$.

Then $n-2\ln(n)>e^k-2(k+1)$

By induction, $e^k-2(k+1)>0$ for all $k\geq2$ (The $n$s for the case $k-1$ can be directly checked.)

The Base case is easy as $e^2-2(2+1)=e^2-6\approx 7.39-6>0$.

For the induction, suppose $e^m-2(m+1)>0$. Then $$e^{m+1}-2(m+2)=ee^{m}-2e(m+1) +2e(m+1)-2m-4$$ Rearranging this, we get $$e(e^{m}-2(m+1)) +2m(e-1)+2(e-2)$$ The first term is positive by the induction hypothesis and the second and third are directly seen to be positive since $e>2$.