I'm taking a course in Set Theory, and I want to prove that finite product of $\omega$ has cardinality $\omega$. I already have that $P(\omega \times \omega) = \omega$. So I can say that using ordinary induction we have that for any $n \in \omega$ we have that $P(\omega_1 \times \ldots \times \omega_n) = \omega$, where $\omega_i=\omega$.
But formally, induction have as base case $n = 0$ and here, I want to use induction with base case $n = 2$. Intuitively it's clear for me that there is no problem, but I can't argue with formal arguments why if the base case is not zero, I can still make induction.
