In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:

Let $\mathbb{Z}_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{Z}_{(p)}$-modules, that is, $\pi,\tau=\mathbb{Z}/p^m$ or $\mathbb{Z}_{(p)}$. Then the stable cohomology groups of the Eilenberg Maclane spaces $\varinjlim H^{i+N}(K(\pi,N);\tau)$ have exponent $p$, i.e. $px=0$ for every element $x$, for $i>0$.

Rudyak says this fact is well-known and references Cartan's Seminaire in 1959. But I have skimmed through the articles in both Seminaire 1958-1959 and Seminaire 1959-1960, yet have not found this fact proved anywhere. Now it is perfectly possible that I have simply overlooked a proposition, since I can barely read French, but I would like to ask if anyone can provide a more specific reference that proves the above fact?

Alternatively, it would be appreciated if anyone can provide a sketch of a proof. Since Rudyak references Cartan Seminaire, it is likely that a proof using Serre's spectral sequence, which is heavily used in the Seminaire, can be constructed. Serre's mod $\mathcal{C}$ theory does not apply here since the groups with exponent $p$ do not form a Serre class. I believe a proof for the case $\pi=\mathbb{Z}/p$ would require a close inspection on how the order of the group dies down. And I don't have any idea how to start with $\pi=\mathbb{Z}_{(p)}$.

Any help is appreciated!