Let $\rho, \sigma \in \mathcal{D}(\mathcal{A})$ with $\text{supp}(\rho) \subseteq \text{supp}(\sigma)$, and spectral decomposition \begin{align*} \rho = \sum_{x}p_x |\psi_x\rangle\langle\psi_x| ~~\text{ and }~~ \sigma = \sum_{x}q_x |\varphi_x\rangle\langle\varphi_x|. \end{align*} For any integer $n$ denote by $y^n :=(y_1, ..., y_n)$, $q_{y^n} := q_{y_1} ... q_{y_n}$, $|\psi_{y^n}\rangle := |\varphi_{y_1}\rangle \otimes ... \otimes |\varphi_{y_n}\rangle$, and for any $\epsilon > 0$ the subspace $T_{n , \epsilon} \subset A^n$ is defined as \begin{align*} T_{n, \epsilon} := \text{span} \left\{ |\Phi_{y^n}\rangle \in A^n: ~ \left| \text{Tr}[\rho \log(\sigma)] - \frac{1}{n} \log q_{y^n} \right| \leq \epsilon \right\}. \end{align*} Finally, denote by $P_{n,\epsilon}$ the projection onto the subspace $T_{n,\epsilon}$ in $A^n$.
(1) Show that $2^{n(Tr[\rho \log(\sigma)] - \epsilon)} P_{n, \epsilon} \leq \sigma^{\otimes n} \leq 2^{n(Tr[ \rho \log(\sigma)] + \epsilon)} P_{n, \epsilon}$.
(2) Show that for any $\epsilon>0: ~ \lim_{n \rightarrow \infty} Tr[\rho^{\otimes n} P_{n,\epsilon}] = 1.$
What I've tried so far:
From the definition of the set $T_{n, \epsilon}$ we obtain $2^{n (Tr[\rho \log\sigma]-\epsilon)} \leq q_{y^n} \leq 2^{n(Tr[\rho \log\sigma]+ \epsilon)}$. Furthermore, we can decompose
$\sigma^{\otimes n} = \sum_{x}q_x |\varphi_x\rangle\langle\varphi_x| = \sum_{x \in T_{n,\epsilon}}q_x |\varphi_x\rangle\langle\varphi_x| + \sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle\varphi_x|$. For those parts of the sum, where x is typical, we can use the derived inequality,
$\sum_{x \in T_{n,\epsilon}} 2^{n (Tr[\rho \log\sigma]-\epsilon)} |\varphi_x\rangle\langle\varphi_x| + \sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle \varphi_x| \leq \sigma^{\otimes n} \leq \sum_{x \in T_{n,\epsilon}} 2^{n (Tr[\rho \log\sigma]+\epsilon)} |\varphi_x\rangle\langle\varphi_x| + \sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle \varphi_x|$
which leads to
$2^{n(Tr[\rho\log\sigma] - \epsilon)} P_{n,\epsilon} + \sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle \varphi_x| \leq \sigma^{\otimes n} \leq 2^{n(Tr[\rho \log\sigma]+\epsilon)} +\sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle \varphi_x| $
For the left inequality we can omit the sum, as all $q_{y^n}$ are positive
$2^{n(Tr[\rho\log\sigma] - \epsilon)} P_{n,\epsilon} \leq \sigma^{\otimes n} \leq 2^{n(Tr[\rho \log\sigma]+\epsilon)} +\sum_{x \in T_{n,\epsilon}^c}q_x |\varphi_x\rangle\langle \varphi_x| $,
but I cannot do the same for the right inequality. I do not see how I can get rid of this term on the RHS.
So, I am stuck with the second inequality of (1) and I do not know how to prove (2), because I do not see any useful connection between the given set and $\rho$. I am grateful for any hints that might help to solve this problem!