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The Cramér model selects integers $n\ge 3$ independently with probability $1/\log n$, and conventionally sets $U_1=0$, $U_2=1$. It is a random model for the distribution of primes, not an assertion of independence for actual prime numbers.

If $P_n$ are the increasing selected integers in the Cramér model, then almost surely ${\mathrm{limsup}}_n(P_{n+1}-P_n)/(\log P_n{)}^2\le 1$. A zero block of length $(1+ϵ)(\log k{)}^2$ after $k$ has summable probability, so the Borel-Cantelli first lemma excludes these blocks eventually.

First, the Cramér model has infinitely many successes almost surely, so every $P_n$ is finite. Indeed, for each fixed $K\ge 3$, independence of the Bernoulli random variables givesThe sum diverges, for example because $1/\log j\ge 1/j$ for $j\ge 3$. Continuity from above of a measure and a countable union bound exclude a final success.

Fix $ϵ>0$ and put $L_k=\lceil (1+ϵ)(\log k{)}^2\rceil$. Let $A_k$ be the event that the whole interval $k+1,\dots ,k+L_k$ consists of failures. By independence and $1-v\le e^{-v}$,Since $L_k=o(k)$, the exponent divided by $\log k$ tends to $-(1+ϵ)$. Thus $\mathbb{P}(A_k)\le k^{-1-ϵ/2}$ for all sufficiently large $k$. The Borel-Cantelli first lemma shows that almost surely every sufficiently large such interval contains a success. Taking $k=P_n$ then gives $P_{n+1}-P_n\le L_{P_n}$ for all sufficiently large $n$.

For each fixed $ϵ$, the limit superior of the normalized gap is therefore at most $1+ϵ$ almost surely. Intersect the probability-one events for $ϵ=1/r$, $r\in \mathbb{N}$, to obtain the Cramér model prime-gap upper bound: