Prove that $\emptyset^*=\{\epsilon\}$

Question

I want to prove that $\emptyset^*=\{\epsilon\}$. I would start by the definition of Kleene star: $$\emptyset^*=\bigcup^\infty_{i=0}\emptyset^i=\emptyset^0+\emptyset^1+...$$ and since $A^0=\{\epsilon\}$ for any set $A$, we have $\emptyset^0=\{\epsilon\}$. Other powers give $\emptyset$ (because $\emptyset \ \cap \ \emptyset=\emptyset$). So the sum is $\{\epsilon\}=\emptyset^*$. Is this correct?

Answer 1

You are right that $\emptyset^n=\emptyset$ for $n\geq1$, but I don't see why that would follow from $\emptyset\cap\emptyset=\emptyset$. Do you mean concatenation there, rather than intersection? Either way I would explain more about how you come to that conclusion.