A little introduction to transfinite induction:
A natural number, in ZF is constructed by ordinal numbers, and such you can continue and construct more, larger numbers, assuming choice we also know for every that every set $S$ there is an unique ordinal $\kappa$ such $|S|=|\kappa|$, so we can define $|S|:=\kappa$.
Each ordinal is exactly one from the following: $0$, $\mu+1$(where $\mu$ is an ordinal), a limit ordinal.
Induction is a way to show take a statement from $0$ to the smallest limit ordinal($\Bbb N=\aleph_0=\omega$), now if we also prove that the statement is true if it is true for every ordinal before the limit ordinal it is also true for the limit ordinal we basically cover all the cases, hence it is true for all the ordinals beneath some ordinal.
This is only a little, the general idea.
Now we use the fact that the ordinals are well ordered, the problem is that we don't know what $|\Bbb R|$ is, we know it is an ordinal but don't know which one. Even more, we don't know how to create a well ordering for $\Bbb R$(or any interval of reals)(we do know such order exists), so we can't use the traditional induction.
That being said, there are other ways, that are not exactly but very close to induction, @Mike Earnest gave two great examples