I am reading the topic on One way ANOVA from the text - Introductory Statistics by Sheldon Ross. To explain the concept of ANOVA, the author has found estimates of $\sigma^2$ using two different approaches and deduced that when the sample means are not equal, the variance found out from the second method is larger in value ( text shown in red in the second image).
How do I prove this is true? Hints quite appreciated! Thank you.
