For a real symmetric matrix , the eigenvalues of it will be positive and smaller or equal to
We could enhance the bound to be 5 actually, but that is pretty much good.
Here I will present a proof linked with Analysis, but the techniques involved would be more or less falling in the categories of high school competitions.
We wish to prove and let .
Observe the RHS: could be rewritten as
By Cauchy-Schwartz, we get an estimate: .
And now we end up proving
To estimate the remaining part, we will estimate it by some positive , a undecided parameters for which holds.
And after plugging in,
after switching sum signs we could get .
We will now prove (*)
pick and so
hence the LHS of (*) will be
and notice that , our new estimate would be It is easy to show hence the conclusion.
I do not know any other proof out there yet, just this proof out there—seems to be a lot of computation.