Use induction to prove that $a^n + b^n \leq (a+b)^n$

Question

I am doing some exercises in proving things and I am stuck on this proof:

$a^n + b^n \leq (a+b)^n$, $a,b > 0$, for every $n > 0$.

I start with $n = 1$: $a^1 + b^1 \leq (a+b)^1$.

Then I assume that $a^n + b^n \leq (a+b)^n$ is true and need to show that $a^{n+1} + b^{n+1} \leq (a+b)^{n+1}$ is true.

And I am stuck from there on.

I know that $a^{n+1} + b^{n+1} \leq (a+b)^n \cdot (a+b)$ but I can't get any further.

Any help please?

Answer 1

$(a+b)^{n+1} = (a+b)^n (a + b) \geq (a^n + b^n)(a + b) = a^{n+1} + ab^n + ba^n + b^{n+1}$.

Can you finish the argument?