Let $a_1=\sqrt{1+2}$,
$a_2=\sqrt{1+2\sqrt{1+3}}$,
$a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}$, …,
$a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}$, … .

Prove that $\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12$.