Prove That $f(n+f(n))=n$

Question

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$

Answer 1

Prove by induction that the $k-1 + f(k-1) < n \le k+ f(k)$ contains one or two elements with image $k$.