This paradox was discovered very soon after people began to grasp the size of the Universe. In the minds of classical
( that is Newtonian) astronomers  the Universe should go on indefinitely in space and time. It was also assumed that the density of luminous bodies, the stars, would be roughly constant everywhere. It is easy to see that at least one of these assumptions is false because the sky is not blindingly ( that is , infinitely ) bright. Suppose the following conditions apply:

assumptions:
1) The average power ( energy per unit time) of a star is P.
2) The average number of stars per unit volume ( the density ) is D.
3) The Universe is infinitely large and eternal.

The question we want to answer is " What is the brightness, or energy per unit time per unit area here on earth?"

We answer this in three steps:
1) What is the brightness due to a single star at a  distance R from the star? See panel (A).
2)  We next calculate the brightness due to many stars of average power P each, that are inside a thin shell of radius R and thickness dR. The number of stars per unit volume is D.  See panel (B).
3) Using the brightness per shell we then add up the contributions from all the shells going out to infinity.

(A) A star of average power P produces a brightness B on a sphere of radius R. What is the brightness?

(B) The earth, at the center, receives energy from stars in the thin shell of radius R and thickness dR. If the density of stars is D and the average power per star is P, find the total brightness due to these stars.

If we find that the answer we get for the total brightness is unphysical, then we know at least one of our assumptions is wrong.!