CW-filtration in K-Theory

Question

I'm just reading Atiyah's paper "Characters and cohomology of finite groups". In §2 he's defining a filtration on $K^*(X)$ by putting $K_p^*(X) = ker \{K^*(X) \rightarrow K^*{(X^{p-1}}) \}$, where $X$ is a finite CW-complex and the map is induced by the inclusion of the $(p-1)$-skeleton. Now he states that $$K^0_{2k-1}(X) = K^0_{2k}(X), \ \ \ \ \ K^1_{2k}(X) = K^1_{2k+1}(X)$$ Why do these equalities hold? As I'm completely lost right now, any help is much appreciated.