I am trying to prove that language $E_{DFA}$ is decidable using multiple executions of $A_{DFA}$ (not using the proof in Sipser's book "Introduction to the Theory of Computation"). 

Can I just use the given $M$ as a decider?

> $M =$ "On input $\langle B,w \rangle$, where $B$ is a DFA and $w$ is
> an empty string ($L(w)=\emptyset$):
> 
> 1. Mark the start of the DFA as $q_0$.
> 2. Simulate B on input w a finite number of times:
 1. First check if the encoding is correct, if $L(w)=\emptyset$, if not - reject.
 2. Recursively: mark states that can be obtained within a finite $δ$
> operations from any marked states.
> 3. If no accept state marked - accept. else, reject."

Is this correct?

Definitions:
$$\begin{align*}
A_{DFA} &= \left\{ \langle B,w \rangle \mid \text{$B$ is a DFA that accepts input string $w$} \right\} \\
E_{DFA} &= \left\{ \langle A \rangle \mid \text{$A$ is a DFA and $L\left(A\right) = \varnothing$} \right\}
\end{align*}$$

Emphasis: I am trying to prove $E_{DFA}$ by running the proof of $A_{DFA}$ several times (finite), so I thought the correct thing would be to use the input $w$ as empty set.