My math is a little rusty. I wish to formally prove to myself that $$\lfloor (z+1)/c \rfloor \le z$$ where $z \in \mathbb{N}$ and $c \ge 2$ for use in a larger inductive proof. The statement seems self-evident given the constraint $c \ge 2$. Still though, I wish to prove it formally, but I'm not sure how to deal with the floor function.

So, so far what I've done is ignore it and instead try and prove $$(z+1)/c \le z$$ To do this, I've rewritten the inequality moving $z$'s to one side: $$1/(c-1) \le z$$ which again looks to be self evident. But how can I really nail this one down? Am I on the right track at least? Or is there a better method of proof available for this problem?