MathJoy

本讲的前提是： For the time being, all quantities occurring are assumed to be rational numbers. 假设我们所知道的数只有有理数，还不知道无理数的存在。 这里说的null-sequence 是rational null-sequence ，定义如下 继续 the second class is empty的例子请看 https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Nested_intervals_theorem 的Nested intervals theorem部分  

每一个实数都能用有理数去逼近到任意精确的程度，这就是有理数的稠密性。The rational points are dense on the number axis.   

$\int _{0}^{1}xdx=\dfrac {1} {2}$