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