The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $$f(n)=n+\left\lfloor\sqrt{n}\right\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$
Prove that for every natural number $m$ the following sequence contains at least one perfect square $m, f(m), f^2(m),f^3(m),\ldots$.
The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times e.g $f^2 = f\circ f.$
I tried this out using induction but I got stuck and couldn't make it. Please help me with this problem.
