In this note the existing proofs of the fact that a real, symmetric, nxn matrix A has a real eigenvalue and that there exists an orthonormal basis for R” consisting of eigenvectors of A are reviewed. A proof of this fact is then given which differs from the existing proofs in that it uses no results from the theory of self‐adjoint operators on complex inner product spaces or from analysis. This makes it possible to prove the spectral theorem for R” in introductory linear algebra courses where, heretofore, such a proof has not usually been given.