Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that $$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$ for any $k\geq 1$. If we know that $\lambda$ has multiplicity at least $M$, then, evidently, we can say something stronger: $$M\cdot \lambda^{2 k} \leq \textrm{Tr} A^{2 k}.$$

Question: how old is each of these two observations? Are they perhaps as old as matrices (Sylvester, Cayley)? Vaguer question: for how long have these observations been the basis of a common strategy for bounding eigenvalues?