Suppose $G$ is a finite group, and $\{ \chi_1,\chi_2,\cdots,\chi_k \}$ be the complete set of irreducible complex characters of $G$. If $\theta$ is a class function on $G$, i.e. function from $G$ to $\mathbb{C}$ which takes same values on conjugacy classes of $G$,
Question: When can we say that $\theta$ is a character of $G$?
If we take characters $\chi_i$ and $\chi_j$, then their sum $\psi=\chi_i + \chi_j$ is also a character of $G$. Looking in other way, the character $\chi_j$ is difference of two characters, $\psi -\chi_i$. So a difference of two characters may be a character of $G$. So if a class function $\theta$ when expressed as an integral linear combination of $\chi_i$'s, then it will be a character provided $\theta(1)>0$.
But in the online notes on "Group Representation Theory", by Daniel Bump (here), he has proved that the class function (namely $\theta^G$ in Theorem 2.5.1 in the notes) is character by showing that it is non-negative integral linear combination of irreducible representations $\chi_i$'s. Why should we show that it is non-negative integral linear combination?