This is a statement made in Hatcher, Algebraic Topology Chpt 3, Sec 2.
One can compute $H^\star(RP^\infty,Z_2)=Z_2[y]$ with $|y|=1$. Now from cellular cochain complex $C^\star(RP^\infty, Z)\to C^\star(RP^\infty, Z_2)$ obtained by projection map $Z\to Z_2$, one deduces $H^\star(RP^\infty, Z)\to H^\star(RP^\infty, Z_2)$ being injection.(It is easy to see $Z\to Z_p$ always inducing chain complex morphism and a cocycle is pushed to a cocycle under chain complex morphism. Checking injectivity is easy then.) Then one deduces $H^\star(RP^\infty,Z)=Z[x]/(2x)$ with $|x|=2$.
Then the book says "Alternatively, this could be deduced from universal coefficient theorem."
$\textbf{Q1:}$ How do I deduce $Z$ valued cohomology from $Z_2$ valued cohomology via universal coefficient theorem?(It is possible to deduce from $Z_2$ cohomology from $Z$ cohomology under universal coefficient theorem. Normally, I would expect somehow patching all primes $p$ of $Z_p$ valued cohomology to $Z$ cohomology which I do not know it is true in general.)
$\textbf{Q2:}$ Why do I expect $H^\star(RP^\infty,Z)=Z[x]/(2x)$ with $|x|=2$ whereas $H^\star(RP^\infty,Z_2)=Z_2[y]$ with $|y|=1$? By computation, one sees $|x|=2$. I do not see why $|x|=2$ is intuitively obvious though $|y|=1$ is obviously true as there is non-trivial cell complex there which is quotient of sphere but $Z_2$ does not see reflection. What is geometric interpretation of $|x|=2$ here? Certainly $Z$ coefficient will see reflection.