Who is interested can find detailed information at wiki, or here
The problem is known (as you added in your edit) as Olbers' paradox, and was posed already in the mid 1500's, by Johannes Kepler in 1610 and even later by Edmond Halley in the eighteen century, and curiously, even the novelist an poet Edgar Allen Poe anticipated possible explanations as to why the night sky is dark But it was popularized as a paradox by German astronomer Olbers in the nineteenth century.
It says that if the universe is infinite and static, then at any given angle from the Earth the line of sight will end at the surface of a star. An infinitely old universe means that there has been plenty of time for the light from every star that has ever shined to reach our eyes, there should be stars everywhere, and the sky at night should be just as bright as when the Sun is up.
The explanation for why the sky is dark instead of comes from recent discoveries about our universe made since Olber's time.
From what was known up to about the nineteenth century, it seemed seemed very reasonable that the universe was infinitely old and unchanging, and in such a universe, Olber's paradox is a real problem.
We now know however, that the universe is not infinitely old and static, the universe had a beginning: the Big Bang. This has important implications for Olbers Paradox. Because the universe has a finite age, one reason our night sky is dark is that many photons have not had time to reach us
But the BB presents another paradox: it states that the early universe was flooded with photons. At this time the cosmos was truly bright. Given these hot, bright early conditions, shouldn't there be a luminous curtain of light behind every star and galaxy we see?
The fact is, this curtain of light is there, but our eyes cannot see it. Due to the expansion of the universe, the wavelengths of these hot, early photons have been stretched over 1,100 times longer than their original wavelengths, the high-energy backdrop of the early universe, is filled with cool, microwave photons (the CMB), invisible to the human eye.
But, after all Olber was right as was Poe: they thought that the night sky should be as bright the noonday Sun, and in fact it is. What they didn't realize was that the night sky is only dark to us. If they had eyes sensitive to microwaves, there would never have been a paradox. One might argue that also frequencies above the visible spectrum are redshifted and become visible, but one should consider the net balance. And then, of course, we must consider the relevance of the excellent reminder of Kyle to Jim, that most part of radiation is reflected away from us, scattered by dust.
EDIT: I've seen both answer may be acceptable, but what is the main
your friend thinks:
...the fact that in a finite amount of time ..light can travel only a certain, finite, distance, [determines the effect that]
we cannot see, if they actually exist, stars farther than such distance.
(and he implicitly thinkss) that, within this radius, there
aren't enough stars to fill the night sky, and that's why we have
darkness without sun's light.
If you have correctly reported your friend' thesis, then your friend is definitely wrong, because within that radius there are enough stars and we can't see their light because of dust
Your thesis is partially wrong, as it is a minor element of the issue, but that is not the main cause, but I think we can safely conclude that: you win the bet by default