试证明下列不等式：
（1）$\frac{n^2}{x_1+\cdots+x_n} \leqslant \frac{1}{x_1}+\cdots+\frac{1}{x_n}\left(x_k>0, k=1,2, \cdots, n\right)$ ．
（2）$\frac{1}{\frac{\alpha_1}{x_1}+\cdots+\frac{\alpha_n}{x_n}} \leqslant x_1^{\alpha_1} \cdots x_n^{\alpha_n} \leqslant \alpha_1 x_1+\cdots+\alpha_n x_n\left(\alpha_k>0, x_k>0(k=1,2, \cdots, n)\right.$, $\left.\sum_{k=1}^n \alpha_k=1\right)$.
（3）$x_1^{\alpha_1} \cdots x_n^{\alpha_n}+y_1^{\alpha_1} \cdots y_n^{\alpha_n} \leqslant\left(x_1+y_1\right)^{\alpha_1} \cdots\left(x_n+y_n\right)^{\alpha_n}\left(\sum_{k=1}^n \alpha_k=1 ; \alpha_k>0, x_k>0, y_k>0, k=\right.$ $1,2, \cdots, n)$.