A direct proof of the existence of eigenvalues and eigenvectors by Weierstrass's theorem

Van Schaftingen, Jean
(2013) The American Mathematical Monthly — Vol. 120, n° 8, p. 737−742 (2013)

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Abstract
I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit of Argand's proof of the fundamental theorem of algebra. The proof only relies on Weierstrass's theorem, the definition of the inverse of a linear operator and algebraic identities.
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Van Schaftingen, J. (2013). A direct proof of the existence of eigenvalues and eigenvectors by Weierstrass’s theorem. The American Mathematical Monthly, 120(8), 737−742. https://doi.org/10.4169/amer.math.monthly.120.08.741 (Original work published 2013)