NL-NWO 1990 Problem 2

Consider the sequence $a_1=\frac{3}{2}, a_{n+1}=\frac{3a_n^2+4a_n-3}{4a_n^2}.$ $(a)$ Prove that $1<a_n$ and $a_{n+1}<a_n$ for all $n$ . $(b)$ From $(a)$ it follows that $\lim_{n\to\infty}a_n$ exists. Find this limit. $(c)$ Determine $\lim_{n\to\infty}a_1a_2a_3...a_n$ .