Given a Deterministic Finite Automata (DFA) $M_1$, does there always exist a Pushdown Automata (PDA) $M_2$ that accepts the same language as $M_1$? I.e. can any DFA be simulated by a PDA? Intuitively, it makes sense to me that a PDA is more powerful since it has an arbitrary amount of memory and can therefore accept more languages than a DFA, but how could this be formally proven?
I hope this answer will help you to understand the mapping. Any DFA is also a PDA. The state transitions in the DFA are similar for PDA without stack. For every transition you perform in DFA, make the similar transition in PDA and do not push/pull from stack.
For a transition
In DFA : $\delta(q_x,0) \rightarrow q_y$
In stack : $\delta(q_x,0) \rightarrow (q_y,\epsilon)$
For accepting state
In DFA : $\delta(q_x,0) \rightarrow q_f$
In stack : $\delta(q_x,0) \rightarrow (q_f,\epsilon)$