I am reading 3rd Chapter "I-projections" of "Information and Statistics : A Tutorial" by Imre Csiszar and Paul C.Shields.
In theorem 3.2, to prove $\mathcal{L} \cap \mathcal{E}_{Q} \neq \varnothing$, it is using a sequence of linear family $\mathcal{L_n}$.
Here
$$\mathcal{L_n} := \{P:\sum_{a}P(a)f_{i}(a)=(1-\frac{1}{n})\alpha_{i}+\frac{1}{n}\sum_{i=1}^{k}Q(a),a\in \mathbb{A}\},$$
$$\mathcal{L} := \{P:\sum_{a}P(a)f_{i}(a)=\alpha_{i},a\in \mathbb{A}\}, $$
$$\mathcal{E}_{Q} := \{P:P(a)=cQ(a)e^{\{\sum_{i=1}^{k}\theta_{i}f_{i}(a)\}},a\in \mathbb{A}\}$$
Following are the snaps of the relevant pages:
I want to know what is the idea/motivation behind this to consider $\mathcal{L_n}$.