In the Bernoulli scheme, \(p=1 / 2\). Prove that

(a)

\[\begin{gathered} \frac{1}{2 \sqrt{n}} \leqslant P_{2 n}(n) \leqslant \frac{1}{\sqrt{2 n+1}} \\ \lim _{n \rightarrow \infty} \frac{P_{2 n}(n+t h)}{P_{2 n}(n)}=e^{-2^{2}} \end{gathered}\]

if \(\frac{h}{\sqrt{n}}=z \quad(0 \leqslant z<+\infty)\).