unknown existence of bijection

Question

Here on the page 4 in the proof of fact 2.6(a) , I do not follow why there is a bijection $$g:\lambda\to\cal P_{\aleph_0}(p)$$ so that we can fix it. We only require $$|A|\leq\lambda,$$ so $\cal P_{\aleph_0}(p)$ may not be large enough ensuring that we can choose $g$ injective. Perhaps there is some item on that page 4, so that this bijection fixing can be done (i.e. that a bijection $g$ exists)?