# Last Inference in proof of conditional limit theorem

I read about the Conditional Limit Theorem from the book "Elements of Information Theory" by Thomas M. Cover and Joy A. Thomas, second edition, page 371. I can't understand the last inference in the proof of the theorem.

The theorem is formulated as follows: Let $E$ be a closed convex subset of probability distributions and let $Q$ be a distribution not in $E$. Let $X_1,X_2,\dots,X_n$ be discrete random variables drawn i.i.d ~ $Q$. Let $P^*$ achieve $\min_{P \in E}D(P||Q)$. Then: $$\mathrm{Pr}\left(X_1 = a \big\vert P_{X^n} \in E\right) \rightarrow P^*(a)$$ in probability, as $n \rightarrow \infty$.

Note: In the formulation above, $P_{X^n}$ means the "type" of the sequence $X^n = (X_1,X_2,\dots,X_n)$.

The end of the proof contains the following statement:

Thus, $\mathrm{Pr}\left(\left|P_{X^n}(a) - P^*(a)\right| \geq \epsilon \big\vert P_{X^n} \in E\right) \rightarrow 0$ as $n \rightarrow \infty$ (Here |.| is used to denote $L_1$ distance). Alternatively, this can be written as $\mathrm{Pr}\left(X_1 = a \big\vert P_{X^n} \in E\right) \rightarrow P^*(a)$ in probability, $a \in \chi$.

I don't understand how to conclude this last inference (in bold).

First of all, |.| does not denote the $L_1$ distance, but the absolute value. Second, the statement follows from the definition of convergence in probability, see for example page 58 in the same book.
Suppose for $$1 \leq i \leq n$$, $$X_i \sim Q$$ iid, then for every $$k$$ we have $$\Pr [X_k =a | P_{X^n} = p] = p(a)$$.