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.