Beginner - Mathematical induction - help understanding example? 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!
 A: In your next to last line, you have $(1+x)^k,$ which you have assumed two lines above is greater than or equal to $1+kx$.  So it made the substitution, using a $\ge$ sign.  You might look at this answer, which has a detailed explanation of induction.
A: Your question is a great one, it is most welcome here! The answer is that, in a proof by induction, we first check the base case (here, it is $n=1$), and then, assuming the result is true for $n=k$, we prove that the result must also be true for $n=k+1$. In other words, we want to prove that
$$\text{true for }n=k\implies\text{true for }n=k+1$$
Intuitively, this lets us say
$$\begin{align} (\text{base case}) \qquad\qquad\qquad\qquad\qquad\qquad&\text{true for }n=1\qquad\checkmark\\
{\text{true for }n=1,\text{ and }\atop (\text{true for }n=k\implies\text{ true for }n=k+1)}\bigg\}\implies&\text{true for }n=2\qquad\checkmark\\
{\text{true for }n=2,\text{ and }\atop (\text{true for }n=k\implies\text{ true for }n=k+1)}\bigg\}\implies&\text{true for }n=3\qquad\checkmark\\
\vdots\end{align}$$
Thus, when we try to prove that the statement is true for $n=k+1$, i.e.
$$(1+x)^{k+1} ≥ 1 + (k + 1)x,$$
we can use the assumption that the statement is true for $n=k$, i.e.
$$(1+x)^k ≥ 1 + kx.$$
The reason why we have
$$(1+x)^k (1+x) ≥ (1+kx)(1+x)$$
is that we are assuming
$$(1+x)^k ≥ 1 + kx$$
is true, and then we multiply both sides by $(1+x)$.
