So:
$$ (1+x)^n ≥ 1 + nx $$
So he checks for 1, and get:
$$ 1+x ≥ 1+x $$
Next for variable k:
$$ (1+x)^k ≥ 1 + kx $$
Then the book wanna prove:
$$ (1+x)^{k+1} ≥ 1 + (k + 1)x $$
And here is books proof:
$$ (1+x)^{k+1} = (1+x)^k (1+x) ≥ (1+kx)(1+x) $$ $$ = 1+(k+1)x + kx^2 ≥ 1 + (k + 1)x $$
Finished! Well... How did the book get this: $(1+kx)(1+x)$ in that last part? Sorry, I'm so confused. Sorry if this to easy to be here. Thanks for all help helping me understand it!