decidability about NFA accepting an input Let A = {(B,w) | B is an NFA that accepts w}. To prove that N is a decidable language,

M = "on input (B,w) where B is NFA and w is a string, simulate B on w. If B accepts w, accept. Otherwise, reject"

Is this a false proof? The proof from the textbook says that I have to convert NFA into equivalent DFA but I'm not sure why this would be necessary. Is it not possible to simulate NFA directly without conversion?
The textbook probably suggests you convert into a DFA because  it is easier to simulate a DFA than to simulate a NFA and determine whether the DFA accepts the word w.  This is trickier on a NFA (but still possible, if you know how).