Proposition 1 is sort of a tautology. They are assuming that the generative classifier is comparing a linear (more accurately affine) function with a threshold to determine the class of the observation. Consider just their continuous example. They say this (near the bottom of page 2)
Note that this method is equivalent to Normal Discriminant Analysis
assuming diagonal covariance matrices.
Reading around this statement, it is clear that they are assuming the variances to be the same under the two classes, so the classifier just reduces to the Linear Discriminant Analysis, that is,
$$
h_{Gen}(x) = 1\{ \hat \beta^T x + \hat b \ge 0\}
$$
for some estimated vector $\hat \beta$ and scalar $\hat b$. Instead of their $\{T,F\}$, I am using $\{0,1\}$. The discriminative classifer also has this form
$$
h_{Dis}(x) = 1\{ \tilde \beta^T x + \tilde b \ge 0\}
$$
for some estimated vector $\tilde \beta$ and scalar $\tilde b$. By their definition, in the discriminative case, these two estimates are obtained by minimizing the population level ($\infty$ sample size) zero-one loss, that is,
$$
(\tilde \beta, \tilde b) = \arg \min_{\beta, b} \mathbb P\big( h_{\beta, b}(x) \neq y\big)
$$
where $h_{\beta,b}(x) := 1\{ \beta^T x + b \ge 0\}$. Thus, by definition,
$$
\mathbb P\big( h_{\tilde \beta, \tilde b}(x) \neq y) \le \mathbb P\big( h_{\beta, b}(x) \neq y), \quad \forall \beta, b.
$$
In particular,
$$
\mathbb P\big( h_{\tilde \beta, \tilde b}(x) \neq y) \le \mathbb P\big( h_{\hat \beta, \hat b}(x) \neq y),
$$
(where the expectation is only w.r.t. $(x,y)$ not the training data),
that is,
$$\mathbb P\big( h_{Dis}(x) \neq y) \le \mathbb P\big( h_{Gen}(x) \neq y).$$
Q.E.D. (Note that there is no convergence in their Prop. 1, everything is defined and stated for the population, $\infty$-sample-size, case).
In short, for the mathematically inclined, $h_{Dis}$, by their definition is the minimizer of $h \mapsto \mathbb P(h(x) \neq y)$ over the class of functions $\mathcal H = \{h_{\beta, b} :\; \beta \in \mathbb R^d, b \in \mathbb R\}$ and $h_{Gen} \in \mathcal H$ again by definition. Hence, the inequality is an almost tautology.
For the second proposition, I would search using these keywords "generalization bounds for empirical risk minimization" or "lecture notes on statistical learning", etc.