Let $H$ be a $\mathbb R$-Hilbert space, $\Phi,\Phi_n\in\mathfrak L(H)$ be compact and self-adjoint, $I:=\mathbb N\cap[1,\operatorname{rank}\Phi],I_n:=\mathbb N\cap[1,\operatorname{rank}\Phi_n]$ and \begin{align}\Phi&=\sum_{i\in I}\lambda_iP_i,\\\Phi_n&=\sum_{i\in I_n}\lambda^{(n)}_iP^{(n)}_i\end{align} be spectral decompositions of $\Phi,\Phi_n$ ($P$ denoting the orthogonal projection of $H$ onto $\mathcal N(\Phi-\lambda_i)$ and so on) for $n\in\mathbb N$ and $\Sigma_i\subseteq\bigcap_{n\in\mathbb N}I_n$ for $i\in I$.
I would like to show that if $$\lambda^{(n)}_j\xrightarrow{n\to\infty}\lambda_i\;\;\;\text{for all }j\in\Sigma_i\tag1$$ and $$\tilde P^{(n)}_i:=\sum_{j\in\Sigma_j}P^{(n)}_j\xrightarrow{n\to\infty}P_i\tag2,$$ then $$\left\|\Phi-\Phi_n\right\|_{\mathfrak L(H)}\xrightarrow{n\to\infty}0\tag3.$$
I've seen a proof of the claim in the special case $H=\mathbb R^d$, $d\in\mathbb N$, $I=\{1,\ldots,r\},I_n=\{1,\ldots,r_n\}$, $r,r_n\in\mathbb N$:
However, I don't get why the first equality should hold. So, I guess there is some missing assumption or something in the setting needs to be fixed.
Note that, in the general case, \begin{equation}\begin{split}&\left\|\sum_{i\in I}\sum_{j\in\Sigma_i}\lambda^{(n)}_jP^{(n)}_j-\sum_{i\in I}\lambda_iP_i\right\|_{\mathfrak L(H)}\\&\;\;\;\;\;\;\;\;\;\;\;\;\le\sum_{i\in I}\left(\sum_{j\in I}\left|\lambda^{(n)}_j-\lambda_i\right|+\lambda_i\left\|\sum_{j\in\Sigma_i}P^{(n)}_j-P_i\right\|_{\mathfrak L(H)}\right)\end{split}\tag4\end{equation} for all $n\in\mathbb N$.