Hilbert Spaces - an application of the minimax principle. Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, 
$$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N \lambda_i $$
Where $\lambda's$ are the eigenvalues ordered by decreasing order (as promised by the minimax principle).
Now, I have reached a line of inequalities, but am not sure about one of the stages. I'll mark it by $\leq^?$:
Take any $N$. Order $e_1,...,e_N$ such that $\langle Ae_1,e_1 \rangle \geq ... \geq\langle Ae_N,e_N \rangle $.
Then $\forall k$:
$$\langle Ae_k,e_k \rangle = \min_{1,...,k} \langle Ae_i,e_i \rangle \leq^? \min_{x \in span\{e_1,...,e_k\}} \langle Ax, x \rangle \leq \max_{V_k} \min_{x\in V_k} \langle Ax,x \rangle = \lambda_k$$
The last equality taken from the minimax principle. The $V_k$ are linear $k$-dimensional subspaces.
But obviously "$\leq^?$" can't be "$\leq$", since I'm taking minimum over a larger set in the RHS, so the minimal value can only decrease. Can "$\leq^?$" be a "$=$", though?
Also, if the statement is not true, what can I do about the question?
I also think that there may be a solution related to one of my previous questions, but I can't connect the two exactly.
 A: Let $A=\sum_k \lambda_k P_k$ be the orthogonal decomposition of $A$ where $u_k$ are its eigenvectors. Let $P=1-\sum_{k=1}^{N-1} P_k$ be the projection on the orthogonal complement of the space spanned by the first $N-1$ eigenvectors. We then have
$$\begin{align}\langle Ae_i,e_i\rangle&=
\langle \sum_{k=1}^\infty\lambda_kP_k e_i,e_i\rangle\\
&=\langle \sum_{k=1}^{N-1}\lambda_kP_k e_i,e_i\rangle+\langle \sum_{k=N}^\infty\lambda_kP_k e_i,e_i\rangle\\
&\leq \langle \sum_{k=1}^{N-1}\lambda_k P_k e_i,e_i\rangle+\langle \lambda_{N}P e_i,e_i\rangle \\
&=\sum_{k=1}^{N-1}\lambda_k|c_{ki}|^2+\lambda_N\left(1-\sum_{k=1}^{N-1}|c_{ki}|^2\right),
\end{align}$$
where $c_{ki}=\langle u_k,e_i\rangle$. Hence
$$\begin{align}\sum_{i=1}^N\langle Ae_i,e_i\rangle&\leq
\sum_{i=1}^N \left(\sum_{k=1}^{N-1}\lambda_k|c_{ki}|^2+\lambda_N\left(1-\sum_{k=1}^{N-1}|c_{ki}|^2\right)\right)\\
&=\sum_{k=1}^{N-1}\lambda_k\sum_{i=1}^N|c_{ki}|^2+\lambda_N\left(N-\sum_{k=1}^{N-1}\sum_{i=1}^N|c_{ki}|^2\right)\\
&\leq\sum_{k=1}^{N-1}\lambda_k+\lambda_N\left(N-(N-1)\right)\\
&=\sum_{k=1}^{N}\lambda_k,
\end{align}$$
where I used the fact that
$$\sum_{i=1}^N|c_{ki}|^2\leq 1,$$
since $\{e_i\}$ is orthonormal.
