A question in classical and quantum information

Let $$\rho, \sigma \in \mathfrak{D}(A)$$ with $$\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$$ and spectral decomposition $$\rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\psi_{x}\right| \quad \text { and } \quad \sigma=\sum_{y} q_{y}\left|\phi_{y}\right\rangle\left\langle\phi_{y}\right|$$ For any integer $$n$$ denote by $$y^{n}:=\left(y_{1}, \ldots, y_{n}\right), q_{y^{n}}:=q_{y_{1}} \cdots q_{y_{n}},\left|\phi_{y^{n}}\right\rangle:=\left|\phi_{y_{1}}\right\rangle \otimes \cdots \otimes\left|\phi_{y_{n}}\right\rangle,$$ and for any $$\epsilon>0$$ the subspace $$\mathfrak{T}_{n, \epsilon} \subset A^{n}$$ is defined as $$T_{n, \epsilon}:=\operatorname{span}\left\{\left|\phi_{y^{n}}\right\rangle \in A^{n}:\left|\operatorname{Tr}[\rho \log \sigma]-\frac{1}{n} \log \left(q_{y^{n}}\right)\right| \leqslant \epsilon\right\}$$ Finally, denote by $$\Pi_{n, \epsilon}$$ the projection onto the subspace $$\mathfrak{T}_{n, \epsilon}$$ in $$A^{n}$$

Now, I want to show two things. First, I want to show that $$2^{n(\operatorname{Tr}[\rho \log \sigma]-\epsilon)} \Pi_{n, \epsilon} \leqslant \sigma^{\otimes n} \leqslant 2^{n(\operatorname{Tr}[\rho \log \sigma]+\epsilon)} \Pi_{n, \epsilon}$$ And I also want to show that for any $$\epsilon>0$$ $$\lim _{n \rightarrow \infty} \operatorname{Tr}\left[\rho^{\otimes n} \Pi_{n, \epsilon}\right]=1$$

$$\textbf{ATTEMPT}$$: I think I should start as follows. I denoted $$r_{y}:=\left\langle\phi_{y}|\rho| \phi_{y}\right\rangle,$$ and by $$\tilde{Y}$$ the random variable whose alphabet is the same as that of $$Y,$$ but his corresponding distribution is $$\left\{r_{y}\right\} .$$ Then, I want to show that $$\operatorname{Tr}[\rho \log \sigma]=\mathbb{E}\left(\log q_{\tilde{Y}}\right)$$