Hi,

I had this induction problem which i couldn't figure out how to solve.

for all n $\displaystyle \geq 1$ show that: $\displaystyle (1+x)^n \geq 1+nx$

The solution looked like this:

$\displaystyle \begin{array}{lcr}(1+x)^{k+1} = (1+x)(1+x)^k \\

\geq ~(1+x){\color{red}(1+kx)} \\

=~1+kx+x+kx^2\\

\geq~1+kx+x\\

=~1+(k+1)x \end{array}$

$\displaystyle \therefore (1+x)^{k+1} \geq 1+(k+1)x$

How can the stuff marked in red be valid?