This is a nice problem which is very similar to a classic, e.g. from German TSTST 2003. The key observation of the short proof is to note that if all the numbers $a_1,\dots,a_k$ are coprime and contained in $[N,N^2)$, then any number in this interval is divisible by at most one of the $a_i$. But the number of multiples of $a_i$ in the interval is certainly more than $\frac{N^2-N}{a_i}-1$. Hence \[N^2-N >\sum_{i=1}^k \left(\frac{N^2-N}{a_i}-1\right)\]and hence $\sum_{i=1}^k \frac{1}{a_i}<1+\frac{k}{N^2-N} \le 2$ as desired.