Prove that sum of eigenvalues of the inverse of an nxn correlation matrix A is greater than or equal to n

Question

I stuck on this question and here is my thoughts:

So we have a nxn correlation matrix A with eigenvalues: λ_1,λ_2,...,λ_n

1.According to the property of correlation matrix, (λ_1)+(λ_2) + ... + (λ_n) = n

2.According to the property of inverse matrix, Let B = $A^{-1}$, then B has eigenvalues: 1/λ_1,1/λ_2,...,1/λ_n

3.Now the question has been transformed into: Prove that 1/λ_1+1/λ_2+...+1/λ_n $\geq$ λ_1+λ_2+..+λ_n and I stuck here

Probably I can use Cauchy inequality to solve it?

So let vector u = (1,1,...1)^T be a n-dim vector, v = (λ_1,λ_2,...,λ_n)^T, then according to C-S inequality:

u·v = ||u||^2*||v||^2.

So we have (λ_1+λ_2+..+λ_n)^2 $\leq$ n*(λ_1^2+λ_2^2+..+λ_n^2), which means n <= (λ_1^2+λ_2^2+..+λ_n^2).

How should I continue now? Appreciate any help!