I am (somewhat) familiar with proving using induction, but why is proof of Bernoulli's inequality, after assuming validity for P(k), set for P(k+1) as:

$\displaystyle \begin{matrix}

(1+x)(1+x)^k \ge (1+x)(1+kx)

& \iff & (1+x)^{k+1} \ge 1+(k+1)x+kx^2

\end{matrix}.$

I don't understand the right side of inequation, shouldn't it be: $\displaystyle \ge 1+(k+1)x$

I see that the inequality must be strictly multiplied by "(1+x)" onbothsides (in orderto remain the sameinequality), but shouldn't one, strictly following rules ofmathematical induction, insert "k+1" on both sides, and thus get $\displaystyle (1+x)(1+x)^k \ge 1+(k+1)x$