Question 18
Let $\{a_n\}$ be a sequence of positive numbers, and $S_n=a_1+a_2+a_3+\cdots+a_n$. For any positive integer $n$, let $b_n=\dfrac12\left(\dfrac{a_{n+1}}{a_n}+\dfrac{a_n}{a_{n+1}}\right)$. Given that $\dfrac{a_n+2}{2}=\sqrt{2S_n}$ holds for all positive integers $n$, determine the limit $\lim_{n\to\infty}(b_1+b_2+\cdots+b_n-n)$.