PAC learnability given information about the expectation of the training/true error
$\mathcal{H}$ is not necessarily pac learnable.

Lets break down your statement, you have an algorithm $\mathcal{A}$, which given a set of labeled samples $S\subseteq \mathcal{X}\times \mathcal{Y}$, generates a hypothesis in $\mathcal{H}$. You also know that $\mathcal{A}(S)$ achieves a generalization error which is close to its sample error (the closeness is determined by $\epsilon_m$). 

If $\mathcal{A}$ was guaranteeing low sample error, then we were safe. However, this is not necessarily the case, take for example the rather stupid learning algorithm which always outputs some constant $h\in \mathcal{H}$ (and ignores the samples). This algorithm has equal sample/generalization error (so the condition is satisfied), but this obviously does not mean $\mathcal{H}$ is pac learnable (you can implement this algorithm even for $\mathcal{H}$ with infinite VC dimension).