homotopy theory - BU with tensor product H-space structure

I'll write $U(X)=[X,BU_otimes]$. So $U(X)subset K(X)=[X,Ztimes BU]$ is the multiplicatively closed subset corresponding to "virtual bundles of rank 1".
Likewise, I'll write $I(X)=[X,BU]subset K(X)$ for the ideal of "virtual bundles of rank 0".
There's a bijection $amapsto 1+a$ from $I(X)$ to $U(X)$.

Here's the claim: if $X$ is compact (let's say it's a finite CW-complex of dimension $n$), then $I(X)$ is a nilpotent ideal (in fact, $I(X)^{n+1}=0$). This is a generic fact about multiplicative cohomology theories, and it can be proved in a number of ways; you can think about it in terms of the multiplicative properties of the Atiyah-Hirzebruch spectral sequence, for instance.

For such $X$, it is clear that elements of $1+ain U(X)$ are invertible, given by the series $(1+a)^{-1}=1+a+a^2+cdots$ which terminates since $ain I(X)$ and $I(X)$ is nilpotent.

For infinite dimensional $X$, you need to argue a little harder. If $X=lim_{to} X_i$ where the $X_i$ are finite CW-complexes, then there is a surjection $K(X)to lim_{leftarrow} K(X_i)$, which restricts to surjections for $I(X)$ and $U(X)$.
The kernel is a $lim^1$-term. If the $lim^1$-term vanishes, then it's clear that elements of $U(X)$ are invertible, since their images in the $U(X_i)$ are invertible. It's enough to check that $lim^1$-vanishes in the case that $X=BU_otimes$, which is a standard calculation.