Prove $(1+\frac{1}{x})^{x+a}>e\ \forall x>0$ when $a\geq1/2$

Question

Prove $(1+\frac{1}{x})^{x+a}>e\ \forall x>0$ when $a\geq1/2$

I can't do this. Please help.

Answer 1

Hint

prove that $\left(1+{1\over x}\right)^{x+a}$ is decreasing and $\left(1+{1\over x}\right)^{x}$ is increasing for $a\ge 0.5$ as Desmos suggests.