When is the empty word part of $A^+$?
You can look at the length of words.

Any element of $A^+$ is a concatenation

$$ a_1 a_2 \ldots a_n $$

where $a_i \in A$ and $n \geq 1$. The length of such a word is

$$ L(a_1 a_2 \ldots a_n) = L(a_1) + L(a_2) + \ldots + L(a_n) $$

But the length is always nonnegative; for example,

$$ 0 \leq L(a_1) \leq L(a_1) + L(a_2) + \ldots + L(a_n)$$

If you're given that

$$ \epsilon = a_1 a_2 \ldots a_n $$

then putting all of the above together gives

$$ 0 \leq L(a_1) \leq L(\epsilon) = 0 $$

and thus $L(a_1) = 0$, so $a_1 = \epsilon$ and $\epsilon \in A$.



If you were to assume that $\epsilon \notin A$, you could modify the above argument by the fact that $L(a_i)$ must be positive, and so

$$ 0 &lt; L(a_1) \leq L(a_1) + L(a_2) + \ldots + L(a_n) $$

and then the final deduction would be

$$ 0 &lt; L(\epsilon) $$

which is a contradiction.