$$\sum_{j=0}^d (s-1)^j \le s^d $$ for all $ s \ge 1 $ and $ d \ge 0 $, where $ s$ and $d$ both are natural numbers.
I have been trying to do this with induction for a while, I have also tried to use the geometric addition formula:
$$\sum_{j=0}^d s^j = \frac{s^{d+1}-1}{s-1} $$
I tried holding $d$ constant and doing Induction on $s$, I also tried the opposite, holding $s$ constant and doing induction on $d$. But it seems I can't solve the problem.
Any ideas would be much appreciated.
In the question $ 0^0 $ is defined as $1$. The below answer should not be taken into consideration.