Sequence problem regarding convergence from an online contest Let $(x_n)_{n\in \mathbb{N}}$ be a sequence defined by $x_0=1$ and $x_n=x_{n-1}\cdot \big(1-\frac{1}{4n^2}\big)$, $\forall n\geq 1$.
Prove that:
a) $(x_n)_{n\in \mathbb{N}}$ is convergent
b) if $l=\lim_{n\to \infty} x_n$, compute $\lim_{n \to \infty} (\frac{x_n}{l})^n$.
What I did was substitute $n-1,n-2,\ldots,1$ in the recurrence relation and I got that $x_n=\prod_{k=1}^{n} \big(1-\frac{1}{4k^2}\big)$. However, here I got stuck because I don't know how to find this limit. 

a) $\frac{x_{n+1}}{x_n}<1$ as $n$ grows. Also, $x_n>0$. So $x_n$ is a strictly decreasing bounded sequence and as a result it's convergent to $l= \inf\{x_n:n \in N\}$.  
