NFA automata with ϵ moves proof
Use the definition of $L(A)$.

There are several different ones, so let me illustrate with the definition I know. We define a function $\hat\delta$ inductively so that $\hat\delta(q_0,w)$ is the set of all states reachable from $q_0$ by reading the word $w$. We then define $L(A)$ as the set of words $w$ such that $\hat\delta(q_0,w)$ intersects $F$.

Given this definition, we argue as follows. First, the definition of $\hat\delta$ (which I haven't specified) easily implies that $q_0 \in \hat\delta(q_0,\epsilon)$. Since $q_0 \in F$, it follows that $\hat\delta(q_0,\epsilon)$ intersects $F$, and so $\epsilon \in L(A)$.