= Solution
Unless a coefficient group is displayed, use integral <singular cohomology>. The standard <CW complex> structure on <infinite-dimensional real projective space> has one cell in each nonnegative dimension. Its <cellular chain complex> has boundary $d_j=2$ for positive even $j$ and $d_j=0$ for odd $j$. The <cellular cohomology> differential $\delta^j$ is therefore zero for even $j$ and multiplication by two for odd $j$. Consequently
$$
\boxed{H^j(\mathbb{RP}^{\infty};\mathbb Z)=\begin{cases}\mathbb Z,&j=0,\\\mathbb Z/2,&j>0\text{ even},\\0,&j\text{ odd}.\end{cases}}
$$
More generally, for an <abelian group> $A$, the positive odd groups are $A[2]=\{a:2a=0\}$ and the positive even groups are $A/2A$. In particular $H^j(\mathbb{RP}^{\infty};\mathbb Z_2)=\mathbb Z_2$ in every nonnegative degree.
For the required <Bockstein homomorphism>, use the <short exact sequence>
$$
0\longrightarrow\mathbb Z_m\overset{\iota}{\longrightarrow}\mathbb Z_{m^2}\overset{q}{\longrightarrow}\mathbb Z_m\longrightarrow0,
\qquad\iota([a])=[ma],\quad q([b])=[b]\pmod m.
$$
Here $m\geq2$; for $m=1$ all three coefficient groups are zero. Since <singular chains> are free <abelian groups>, applying cochains gives a <short exact sequence> of <cochain complexes>. Its <connecting homomorphism> defines $\beta$ and its <long exact sequence> is precisely the required one, with the other maps induced by $\iota$ and $q$.
Explicitly, represent a class by a <cocycle> $u\in C^n(X;\mathbb Z_m)$ and choose a lift $\widetilde u\in C^n(X;\mathbb Z_{m^2})$. Since $q(\delta\widetilde u)=0$, there is a unique cochain $v$ with $\iota(v)=\delta\widetilde u$. Injectivity of $\iota$ and $\delta^2=0$ show $\delta v=0$. Define
$$
\boxed{\beta([u])=[v].}
$$
Changing the lift by $\iota(w)$ changes $v$ by the <coboundary> $\delta w$. Changing the representative $u$ by a <coboundary> can be lifted by a coboundary as well and leaves the resulting class unchanged. Thus this is a well-defined <group homomorphism>, and the standard cochain lifting argument gives exactness.
Compute the <Bockstein homomorphism> on <infinite-dimensional real projective space> using its <cellular cohomology> complex. A generator with coefficients $\mathbb Z_2$ is represented by $1$ in degree $n$, lifted to $1\in\mathbb Z_4$. Its coboundary is $0$ for even $n$ and $2$ for odd $n$. Dividing via $\iota(1)=2$ gives
$$
\boxed{\beta=0\text{ for even }n,\qquad\beta=\mathrm{id}_{\mathbb Z_2}\text{ for odd }n.}
$$
Thus the odd-degree maps are \b[isomorphisms]. The comparison between <cellular cohomology> and <singular cohomology> is natural with respect to coefficient maps, so this computes the same <connecting homomorphism> constructed above.
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