Induction proof with numerator and denominator

Question

I am stuck during the induction process where everything will fall into place if I show: $\frac{x+1}{1-mx} \leq \frac{1}{1 - (m+1)x}$ for any $m \in \mathbb{N}$ and any $x \in (0, \frac{1}{m+1})$.

Any suggestions?

Answer 1

The given inequality is equivalent to

$$1-(m+1)x+x-(m+1)x^2\le 1-mx$$

cancelling out $1-mx$, we get

$$-(m+1)x^2\le 0,$$ which is clearly true.