Doubt in the proof of Blichfeldt's Theorem given in Wiki Suppose we are drawing some patches in a paper of size $1\times 1$. In the end, we discover that the sum of the area of all the patches is $>n$. Prove that there must exist at-least one point such that it lies in at-least $n+1$ of the patches.
This was a part in the proof of Blitchfeldt Theorem. The proof given in the link does not really seem convincing to me. The idea of contributions of points towards the area doesn't seem rigorous or at-least needs more justification in my opinion. So I was trying a "set-theoretic" approach, but could not get far.
Progress: When $n=1$, if no two patches have a area in common, then they all lie inside a square of area 1 and are disjoint, hence their total area can not exceed $1$. I could also show it for the case $n=2$. But could not for $n>2$. Any help would be appreciated.

Imagine painting each patch with an $\epsilon$ thick layer of paint. When a point is painted twice, the paint stacks in thickness. The total volume of paint is $\epsilon$ times the sum of the areas of the patches, so the volume is more than $\epsilon n$.
If you suppose by way of contradiction that no point lies in $n+1$ patches, then the maximum thickness of paint would be $n\cdot \epsilon$, so all of the paint would fit inside a $1\times 1\times (\epsilon n)$ box. This would mean the volume is also at most $\epsilon n$, which is a contradiction.
All of this can be formulated rigorously.
