Proof by induction problem explanation

[xeon123]

In this link (https://www.physicsforums.com/showthread.php?t=523874 ) there's an example of proof by induction.

Somewhere in the explanation there's the phrase:

Because x+1≥0, we can multiplicate both sides by x+1.

Why they decided to multiply both sides by x+1?

 

[eumyang]

It was assumed that
(1) $(1+x)^n\geq 1+nx$

And you want to prove that
(2) $(1+x)^{n+1}\geq 1+(n+1)x$

By multiplying both sides of (1) by (1+x), you'll get the LHS of the resulting inequality to match the LHS of (2).

$a^m a = a^m a^1 = a^{m+1}$ by the product property of exponents, so
$(1+x)^n (1+x) = (1+x)^{n+1}$.

 

[xeon123]

So, the equation will be: $(1+x)^{n+1} \geq (1+x)(1+nx)$?

From the RHS I can't get 1+(n+1)x, or can I?

 

[HallsofIvy]

Well, what do you get when you multiply on the right?

 

[xeon123]

$(1+x)(1+nx)$ is equal to $1+(n+1)x$?