单愿你能看懂。

Rigorous proof by: gyro

Let \int denote integral, x denote time(year), N(x) denote the number of people at age x. Each year there are -dN/dx=-N' people die at age x. The total number of people die in a year is -\int N'dx

The average life span=\int N'xdx/\int N' dx

The average death rate is -\int N'dx/\int N dx

Integration by part,

\int N'xdx=-\int N dx

So life span x death rate = 1

This conclusion holds regardless of the variation of the death rate with age.