## 证明

$$\frac{\frac{\frac{\frac{\frac{x+L_{1}a_{1}}{L_{1}+1}+L_{2}a_{2}}{L_{2}+1}…+L_{l}a_{l}}{L_{l}+1}+h_{1}}{2}+h_{2}}{2}….$$

$$\frac{\frac{\frac{\frac{x+L_{2}a_{2}}{L_{2}+1}…+L_{l}a_{l}-a_{l+1}}{L_{l}}+a_{l+1}}{2}+h_{1}}{2}….$$

$$\frac{\frac{\frac{x+L_{1}a_{1}}{L_{1}+1}+L_{2}a_{2}}{L_{2}+1}…+L_{l}a_{l}}{L_{l}+1}$$

$$\frac{\frac{\frac{x+L_{2}a_{2}}{L_{2}+1}…+L_{l}a_{l}-a_{l+1}}{L_{l}}+a_{l+1}}{2}$$

$$\frac{x}{\Pi_{i=1}^{l}(L_{i}+1)}+\sum_{i=1}^{l}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}{(L_{j}+1})}$$
$$(L_{1}+1)\frac{x}{\Pi_{i=1}^{l}(L_{i}+1)\frac{2L_{l}}{L_{l}+1}}+\sum_{i=1}^{l}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}(L_{j}+1)\frac{2L_{l}}{L_{l}+1}}+a_{l+1}\frac{L_{l}-1}{2L_{l}}$$

$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq(\frac{(L_{l}+1)(L_{1}+1)}{2L_{l}}-1)\frac{x}{\Pi_{i=1}^{l}(L_{i}+1)}+\frac{1-L_{l}}{2L_{l}}\sum_{i=1}^{l}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}{(L_{j}+1})}+a_{l+1}\frac{L_{l}-1}{2L_{l}}$$

$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq\frac{1-L_{l}}{2L_{l}}\sum_{i=1}^{l}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}{(L_{j}+1})}+a_{l+1}\frac{L_{l}-1}{2L_{l}}$$
$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq\frac{L_{l}-1}{2L_{l}}(a_{l}-\sum_{i=1}^{l}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}{(L_{j}+1})})$$

$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq\frac{L_{l}-1}{2L_{l}}(\frac{1}{L_{l}+1}a_{l}-\sum_{i=1}^{l-1}\frac{L_{i}a_{i}}{\Pi_{j=i}^{j<=l}{(L_{j}+1})})+\frac{L_{l}-1}{2L_{l}}\Delta$$

$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq\frac{L_{l}-1}{2L_{l}}(\frac{a_{1}}{\Pi_{i=1}^{l}{(L_{i}+1})})+\frac{L_{l}-1}{2L_{l}}\Delta$$

$$\frac{L_{1}a_{1}}{\Pi_{i=1}^{l}(L_{i}+1)}\geq\frac{L_{l}-1}{2L_{l}}\Delta$$