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Source. Equation (2), printed p. 85 (PDF p. 1). The paper attributes the upper bound to its reference [3]: P. Erdős, Számelméleti megjegyzések IV, Matematikai Lapok 13 (1962), 241–243. That original proof is not reconstructed in this source unit.

Imported statement. A disjoint system with $k\ge1$ distinct proper moduli $2\le n_1<\cdots<n_k$ satisfies

$$\sum_{i=1}^k\frac1{n_i}\le1-2^{-k}. \tag{1}$$

The weak inequality is what the PDF prints. Modulus one must be excluded: its single progression would violate (1).

Sharpness example. For $1\le i\le k$, take the class $2^{i-1}\pmod{2^i}$. A member has exact 2-adic valuation $i-1$, so these classes are pairwise disjoint. Their distinct proper moduli have reciprocal sum $\sum_{i=1}^k2^{-i}=1-2^{-k}$. Thus equality is attained.

This verifies the source's sharpness remark only. It is not a proof of the upper bound for arbitrary systems. Equation (1) is not an input to the complete proof of $f(x)=o(x)$ on Theorem 1. The source's separate cited impossibility of a distinct-modulus disjoint covering is likewise an external historical result.

Bears on. The reciprocal-sum background for Problem 1190. This finite-$k$ bound is not the later optimized tail estimate in that problem.