X据/span>}据/span>据据/span>1据/span>。自从据/p>
⌊据/span>X据/span>⌋据/span>≤.据/span>⌊据/span>X据/span>+据/span>N据/span>1据/span>⌋据/span>≤.据/span>⋯据/span>≤.据/span>⌊据/span>X据/span>+据/span>N据/span>N据/span>-据/span>1据/span>⌋据/span>≤.据/span>⌊据/span>X据/span>⌋据/span>+据/span>1据/span>那据/span>
存在一个据span class="katex">
一世据/span>这样据/p>
⌊据/span>X据/span>⌋据/span>=据/span>⌊据/span>X据/span>+据/span>N据/span>1据/span>⌋据/span>=据/span>⋯据/span>=据/span>⌊据/span>X据/span>+据/span>N据/span>一世据/span>-据/span>1据/span>⌋据/span>
和据/p>
⌊据/span>X据/span>+据/span>N据/span>一世据/span>⌋据/span>=据/span>⌊据/span>X据/span>+据/span>N据/span>一世据/span>+据/span>1据/span>⌋据/span>=据/span>⋯据/span>=据/span>⌊据/span>X据/span>+据/span>N据/span>N据/span>-据/span>1据/span>⌋据/span>=据/span>⌊据/span>X据/span>⌋据/span>+据/span>1据/span>。据/span>
因此据/p>
⌊据/span>X据/span>⌋据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>1据/span>⌋据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>2据/span>⌋据/span>+据/span>⋯据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>N据/span>-据/span>1据/span>⌋据/span>=据/span>一世据/span>⌊据/span>X据/span>⌋据/span>+据/span>(据/span>N据/span>-据/span>一世据/span>)据/span>(据/span>⌊据/span>X据/span>⌋据/span>+据/span>1据/span>)据/span>=据/span>N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>-据/span>一世据/span>。据/span>
现在注意,据span class="katex">
一世据/span>满足据/p>
{据/span>X据/span>}据/span>+据/span>N据/span>一世据/span>-据/span>1据/span>据据/span>1据/span>
和据/p>
{据/span>X据/span>}据/span>+据/span>N据/span>一世据/span>≥据/span>1据/span>。据/span>
这两个不平等加在一起给了我们据/p>
N据/span>N据/span>-据/span>一世据/span>≤.据/span>{据/span>X据/span>}据/span>据据/span>N据/span>N据/span>-据/span>一世据/span>+据/span>1据/span>。据/span>
利用这一点,据span class="katex">
N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>-据/span>一世据/span>≤.据/span>N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>{据/span>X据/span>}据/span>=据/span>N据/span>X据/span>据据/span>N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>-据/span>一世据/span>+据/span>1据/span>那据/span>这意味着据span class="katex">
⌊据/span>N据/span>X据/span>⌋据/span>=据/span>N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>-据/span>一世据/span>。据/p>
因此据/p>
⌊据/span>X据/span>⌋据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>1据/span>⌋据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>2据/span>⌋据/span>+据/span>⋯据/span>+据/span>⌊据/span>X据/span>+据/span>N据/span>N据/span>-据/span>1据/span>⌋据/span>=据/span>一世据/span>⌊据/span>X据/span>⌋据/span>+据/span>(据/span>N据/span>-据/span>一世据/span>)据/span>(据/span>⌊据/span>X据/span>⌋据/span>+据/span>1据/span>)据/span>=据/span>N据/span>⌊据/span>X据/span>⌋据/span>+据/span>N据/span>-据/span>一世据/span>=据/span>⌊据/span>N据/span>X据/span>⌋据/span>那据/span>
这完成了证明。据span class="katex">
□据/span>