The Pseudoforest analogue for the Strong Nine Dragon Tree Conjecture is True

We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},ldots,C_{k+1}$ such that there is an $i$ such that for every connected component $C$ of $C_{i}$, we have that $e(C) leq d$.