How do the geometric considerations ($B imes N$) of an infinite, static universe resolve the paradox mathematically?

The number of stars increases proportionally with the volume ($N imes d^3$), perfectly counteracting the dimming effect of distance ($B imes 1/d^2$).

In a hypothetical infinite and static universe, while the apparent brightness ($B$) of an individual source dims according to the inverse-square law ($B imes 1/d^2$), the volume containing those sources increases with the cube of the distance ($V imes d^3$), meaning the number of stars ($N$) increases proportionally to $d^3$. Mathematically, these two effects cancel each other out, predicting that the sky brightness remains constant regardless of distance, leading directly to the paradox that the sky should blaze brightly.