\textbf{1}
(1) For a positive integer $k$, let $A_k = \displaystyle\int_{\sqrt{k\pi}}^{\sqrt{(k+1)\pi}} |\sin(x^2)|\,dx$. Prove that the following inequality holds:
$$\frac{1}{\sqrt{(k+1)\pi}} \leqq A_k \leqq \frac{1}{\sqrt{k\pi}}$$
(2) For a positive integer $n$, let $B_n = \dfrac{1}{\sqrt{n}}\displaystyle\int_{\sqrt{n\pi}}^{\sqrt{2n\pi}} |\sin(x^2)|\,dx$. Find the limit $\displaystyle\lim_{n\to\infty} B_n$.
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