= Solution
Use the <short exact sequence> of coefficient groups
$$
0\longrightarrow\mathbb F_p\xrightarrow{\ j\ }\mathbb Z/p^2\xrightarrow{\ r\ }\mathbb F_p\longrightarrow0,\qquad j(\bar a)=\overline{pa},\quad r(\bar b)=\bar b\pmod p.
$$
The groups of <singular chains> are free, so applying the cochain construction gives a <short exact sequence> of <cochain complexes>. Its <long exact sequence in cohomology> has the required terms and reduction map $r_*$. Define the <Bockstein homomorphism> to be its <connecting homomorphism>.
Explicitly, represent $x$ by a mod-$p$ <cocycle> $a$ and lift it to an integral <singular cochain> $A$. Since $a$ is a <cocycle>, $\delta A$ is divisible by $p$. Then
$$
\boxed{\beta(x)=\left[\frac{\delta A}{p}\bmod p\right].}
$$
This is exactly the lift-and-<coboundary> construction for the coefficient sequence above: the mod-$p^2$ <coboundary> of $A$ lies in the image of $j$. The quotient cochain is a <cocycle>, since $\delta^2A=0$ and integral cochains have no $p$-torsion. Replacing $A$ by another lift adds $pC$ and changes the quotient by $\delta C$ modulo $p$; replacing the representative by a <coboundary> likewise leaves its <cohomology> class unchanged. Thus the definition is independent of choices.
A degree-zero <cocycle> is a function on points constant on each path component. Choose an integer representative for its value on each component. This produces an integral zero-<cocycle> lift, with zero <coboundary>. Hence \b[the Bockstein vanishes on $H^0(X;\mathbb F_p)$ for every space].
For a nonzero example in every positive degree $i$, take the <Moore space> $X=S^i\cup_p e^{i+1}$, attaching the cell by a map of <mapping degree> $p$. Its positive-degree integral <cellular cochain complex> has the differential $\mathbb Z\xrightarrow{p}\mathbb Z$ from degree $i$ to $i+1$; when $i=1$, the preceding differential from degree zero is zero. Modulo $p$ the displayed differential vanishes, and $H^i(X;\mathbb F_p)=H^{i+1}(X;\mathbb F_p)=\mathbb F_p$. Lift the degree-$i$ generator to the cellular cochain with value one. Its <coboundary> has value $p$, so dividing by $p$ gives the generator in degree $i+1$. Naturality of cellular and singular <cohomology> with coefficient sequences identifies this with the <Bockstein on a cyclic Moore space>. Therefore \b[$\beta$ is nonzero, indeed an isomorphism, in the requested degree for every prime $p$].
Finally take integral lifts $A,B$ of cocycles representing $x\in H^i(X;\mathbb F_p)$ and $y\in H^j(X;\mathbb F_p)$. Write $\delta A=pA_1$ and $\delta B=pB_1$. The cochain <cup product> obeys the graded Leibniz identity, so
$$
\delta(A\smile B)=p\bigl(A_1\smile B+(-1)^i A\smile B_1\bigr).
$$
Divide by $p$, reduce modulo $p$ and pass to <cohomology>. This proves the <Bockstein derivation rule>
$$
\boxed{\beta(xy)=(\beta x)y+(-1)^i x(\beta y).}
$$
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