Please I need help with this question:
$E(x)= \left \{ \left(1+ \frac{x}{n} \right)^n : n \in \mathbb{N} \right \}$. Let $a(x) = \sup E(x)$ (least upper bound).
($1$) Prove that $a(x) < a(y)$ if $0 < x < y$.
(2) Prove that $a(x)a(y) \leq \left( a \left( \frac{x+y}{2} \right) \right) ^ 2$.
(3) Prove that $a(x + y) = a(x) \cdot a(y)$.
I already proved that $E(x)$ has no largest element and that it's bounded above. I also proved the first two parts. And I know that E(x) is bounded by e^x and thus supE(x) = e^x = a(x), so a(y) = e^y. Can you please help me in proving the third part?
Thank you.