函数列f在点集I上一致收敛,其平方在同一点集I上不一致收敛,举个实例.

fn(x)＝1/x+1/n,0显然fn(x)在(0,1)一致收敛于f(x)＝1/x.
但fn^2(x)＝1/x^2+2/(nx)+1/n^2,f^2(x)＝1/x^2,
此时fn^2(x)－f^2(x)＝2/(nx)+1/n^2,
fn^2(x)－f^2(x)在x＝1/n的函数值为2,
因此不一致收敛.