We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.
Mercer, A. McD., & Mercer, P. R. (2000). Cauchy’s interlace theorem and lower bounds for the spectral radius. International Journal of Mathematics and Mathematical Sciences, 23(8), 563–566. https://doi.org/10.1155/s016117120000257x