I want to show that a graph G that is a union of k forests has a chromatic number of at most 2k. I have narrowed my problem down to trying to show that any graph G that is a union of n trees has a vertex of degree at most 2n-1, thus I can show that G is 2k-1 degenerate and prove this theorem by using a known inequality (Chromatic Number $\leq$ Degeneracy of G + 1). that would suffice because any subgraph of G would also be a union of k forests or less, so G would indeed be 2k-1 degenerate.
However I've had difficulties proving this general fact about graphs made of unions of forests and would appreciate some guidance. I'm not sure what the best approach would be, some traits of trees/forests or something else, was unable to make progress via induction.
Thanks a million!