The statement in the title is true in a number of ways, but here is a very simple argument.



Let us say we have decisively "falsified" a theory: rocks float on water, perhaps. So, per Popper, we now abandon that theory, and come up with a new bold conjecture.



Well, this abandonment depends entirely on an inductive inference! In fact, it depends on the principle of "conservative induction" being true: more of the same. Unless that is the case, why should the theory having been falsified today having anything whatsoever to do with whether we should employ it tomorrow?



Imagine that, instead, the principle of revolutionary induction is true: time for a change. In that case, the fact that our theory was falsified today should give us great hope for the theory tomorrow. And if that principle is true, then we should abandon all of our theories that have not been falsified right away!



So, falsification, far from solving the "problem" of induction by eliminating it, actually depends on conservative induction being justified. And this is not the only way in which falsification depends on induction, just the simplest that I have thought of.



These arguments are well known in the literature, and considered decisive by almost all professionals in the philosophy of science, which is why there are only a handful of Popperians left in the field. (I'd be surprised if, in the top 100 departments in the field, there are a total of more than five Popperians -- not five Popperian departments, but five individual Popperians.) And this is certainly not because Popper is a neglected figure or was out of the mainstream: no, Popper and Hempel were the two giant figures of the 50s and 60s. These ideas were carefully considered, and found to be wanting.