Terrence Tao formulates strong induction as follows: "(Strong principle of induction). Let $m_o$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number 𝑚. Suppose that for each $m \geq m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 \leq m' \lt m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true since in this case, the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m \geq m_0$. "
Since we are assuming that $P(m')$ is true for all $m_0 \leq m' \lt m$. Where in this statement does it suggest to prove the base case first? To me all its suggesting is to simply suppose $P(m')$ is true for all $m_0 \leq m' \lt m$ without proving a base case?