vector bundles - Products and the skeletal filtration in K-theory

Given a finite CW complex X, there is a filtration of the topological K-theory of X given by setting $K_n(X) = ker left(K(X) to K(X^{(n-1)})right)$, where $X^{(n-1)}$ is the (n-1)-skeleton of X. (The choice of indexing here is from Atiyah-Hirzebruch.)

My question is:

How does this filtration interact with the external product $K(X)times K(Y)to K(X times Y)$? I believe the answer should be that $K_n (X) cdot K_m (Y) subset K_{n+m} (Xtimes Y)$.

Just to be clear, and to set notation, this external product is the one induced by sending a pair of vector bundles $Vto X$ and $Wto Y$ to the external tensor product, which I'll write $Vwidetilde{otimes} W = pi_1^* V otimes pi_2^* W to Xtimes Y$.

Of course, if $Vin K_n (X)$ and $W in K_m (Y)$, then $Vwidetilde{otimes} W$ restricts to zero in both $K(X^{(n-1)} times Y)$ and $K(X times Y^{(m-1)})$, and $(Xtimes Y)^{(n+m-1)}$ is contained in the union of these two subsets. Is there some way to deduce from this information that the class $Vwidetilde{otimes} W$ is actually trivial in $K((Xtimes Y)^{(n+m-1)})$?

Here's the reason I'm asking (which is really a second question, I guess).
In Characters and Cohomology Theories, Atiyah states (without comment) that for the internal product $K(X)times K(X)to K(X)$, one has $K_n (X) cdot K_m (X) subset K_{n+m} (X)$. In Atiyah-Hirzebruch, they state this formula and say that it "admits a straighforward proof."

I thought I remembered that the straighforward proof was the following:

1. Show that the external product satisfies $K_n (X) cdot K_m (Y) subset K_{n+m} (Xtimes Y)$




2. Observe that if $f:Xto Xtimes X$ is a cellular approximation to the diagonal $Xto Xtimes X$, then $f(X^{(n+m-1)}) subset (Xtimes X)^{(n+m-1)}$. So for any $V, Win K(X)$, we have $Votimes W = f^*(Vwidetilde{otimes} W)$, and if $Vin K_n (X)$ and $Win K_m (X)$, it then follows from 1. that $Votimes Win K_{n+m} (X)$.

Am I barking up the wrong tree here?

Presumably these questions will turn out to have an easy answer, but I've been thinking about them for a while now and haven't gotten any further. Any suggestions or references would be great! I haven't found any sources other than the two mentioned above that talk about the relation between skeleta and products, and neither of these sources mentions case of external products.