I'm reading through Larry Wasserman's book, All of Statistics, and currently about p-values (page 187). Let me first introduce some definitions (I quote):
Definition 1 The power function of a test with rejection region $R$ is defined by $$\beta(\theta)=P_{\theta}(X\in R)$$ The size of a test is defined to be $$\alpha = \sup_{\theta\in\Theta_0}\beta(\theta)$$ A test is said to have level $\alpha$ if its size is less than or equal $\alpha$.
This basically says that $\alpha$, the size is the "biggest" probability of an error of type I. The $p$-value is then defined via (I quote)
Definition 2 Suppose that for every $\alpha\in(0,1)$ we have a size $\alpha$ test with rejection region $R_\alpha$. Then, $$p\text{-value}=\inf\{\alpha:T(X^n)\in R_\alpha\}$$ where $X^n=(X_1,\dots,X_n)$.
For me this means: given a specific $\alpha$ there is a test and rejection region $R_\alpha$ so that $\alpha=\sup_{\theta\in\Theta_{0}(\alpha)}P_\theta(T(X^n)\in R_\alpha)$. For the $p$-value I simply take then the smallest of all these $\alpha$.
Question 1 If this would be the case, then I could clearly choose $\alpha = \epsilon$ for arbitrarily small $\epsilon$. What is my wrong interpretation of definition 2, i.e. what does it exactly mean?
Now Wasserman continuous and states a theorem to have an "equivalent" definition of $p$-value with which I'm familiar (I quote):
Theorem Suppose that the size $\alpha$ test is of the form $$\text{reject } H_0 \iff T(X^n)\ge c_\alpha$$ Then, $$p\text{-value} = \sup_{\theta\in\Theta_0}P_{\theta}(T(X^n)\ge T(x^n))$$ where $x^n$ is the observed value of $X^n$.
So here is my second question:
Question 2 How can I actually prove this theorem? Maybe it's due to my misunderstanding of the definition of the $p$-value, but I can't figure it out.