In this paper we approximate a Schrödinger operator on by Jacobi operators on to provide new proofs of sharp Lieb-Thirring inequalities for the powers and . To this end we first investigate spectral inequalities for Jacobi operators. Using the commutation method, we present a new, direct proof of a sharp inequality corresponding to a Lieb-Thirring inequality for the power on . We also introduce inequalities for higher powers of the eigenvalues as well as for matrix-valued potentials and compare our results to previously established bounds.