In my textbook, it presents the theorem, "A language L is accepted by some DFA if and only if l is accepted by some NFA". My textbook explains that the "if" portion of the proof is given by the subset construction and theorem if $D=(Q_D, \sigma, \delta_D, {q0}, F_D)$ is the DFA constructed from NFA $N=(Q_N, \sigma, \delta_N, {q0}, F_N)$ by the subset construction, then L(N) = L(D).
However, it leaves it up to the reader to prove the "only if" portion - essentitally, if $\hat\delta_D(q,a) = p$ then $\hat\delta_N(q,a) = {p}$. It suggests that we mu prove this by induction on $|w|$.
I understand how to complete the basis part of this inductive proof, I think.
If w is the empty string, then $$\delta_D(q,\epsilon) = q$$ and $$\delta_N(q,\epsilon) = {q}$$ by definition of the extended transition function.
However, I'm lost as to how to finish the proof. I've come up the starts of something, but I don't know if I'm even in the right direction:
Assume the statement is true for strings shorter than y and break y into $xa$ where $a$ is last symbol of $y$ and x is the string of symbols before a.
$\delta_N(q_0, y) = \bigcup\limits_{i=1}\delta_n(p_i, a)$
$\delta_D(q_0, y) = \delta(\hat\delta(q_), x), a)$