How did people arrive that only 3.14 defects are allowed for million opportunities for six sigma? Is there any mathematical/statistical proof for this? Likewise for 3sigma,4sigma and 5sigma?

This can be understood with the help of Normal Distribution. 

 

Consider the Upper Specification Limit (USL) as Mean + n * S.D.

 

Consider the Lower Specification Limit (USL) as Mean - n * S.D.

 

The area between  the USL & LSL is the defect free area and the outliers are the defects.

 

In case of Six Sigma, n = 6.

 

The area under the curve is 0.9999966.

Outside the curve, we are left with .0000034, this means 3.4 defects per million.

mathematical explanation is available.. Pls refer to Z tables...