Max-Min Principle with an example on intervals

Question

I have a few difficulties to understand the max-min principle (Intuitive understanding of Maximin Principle). Is there anyone could explain this theorem in using $[a,b] \subset [a',b']$? I know that the eigenvalues related to these interval is one the form $\lambda_n = \frac{n^2 \pi^2}{l^2}$ ($l$ is the lenght of an interval) with the eigenfunction $u_n= \sin (\frac{n \pi}{l} x)$.