Growth of Exponential Expressions

If $\alpha$ and $\beta$ are positive quantities, and $a>\beta$, show that ${\left(1+\frac{1}{\alpha }\right)}^{\alpha }>{\left(1+\frac{1}{\beta }\right)}^{\beta }$. Hence show that if $n>1$ the value of ${\left(1+\frac{1}{n}\right)}^n$ lies between $2$ and $2.718\dots$

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIX. Inequalities, Examples XIX.b, No. 4