Is it possible to show only by induction and without the binomial theorem that $(1+\frac{1}{n+1})^{n+1} - (1+\frac{1}{n})^{n} < 1$?

Question

If I could show that than it would be equivalent to say that there is no natural number in the interval $ (2,(1+\frac{1}{n})^{n})] \Rightarrow (1+\frac{1}{n})^{n} < 3 $

Answer 1

Hint:

$\Bigl(1+\dfrac1{n+1}\Bigr)^{\!n+1}<\mathrm e$. On the other hand, $\Bigl(1+\dfrac1n\Bigr)^{\!n}>1+n\cdot\dfrac1n=2$ (Bernoulli's inequality), so $$\Bigl(1+\dfrac1{n+1}\Bigr)^{\!n+1}-\Bigl(1+\dfrac1n\Bigr)^{\!n}<\mathrm e-2.$$