We prove the Courant-Fischer Min-Max Theorem. For a symmetric matrix A, we can arrange the real eigenvalues in increasing order. We know from Rayleigh's Theorem (https://youtu.be/r8KhlywcWUE) that the maximal eigenvalue can be achieved by maximizing the Rayleigh quotient. We sharpen this result by finding any of the eigenvalues by looking subspaces of increasing dimension.
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