The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $$f(n) = n + \lfloor \sqrt{n} \rfloor$$ where $\lfloor k \rfloor$ denotes the nearest integer smaller than or equal to $k$. For example, $\lfloor \pi \rfloor = 3$, $\lfloor 4 \rfloor = 4$. 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., $f^{2} = f \circ f$.
The domain of a function $f$ is the set of natural numbers. The function is defined as follows:
$$f(n) = n + \lfloor \sqrt{n} \rfloor$$
where $\lfloor k \rfloor$ denotes the nearest integer smaller than or equal to $k$. For example, $\lfloor \pi \rfloor = 3$, $\lfloor 4 \rfloor = 4$. 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., $f^{2} = f \circ f$.