No, the night sky will not eventually end up completely black. It is true that the universe is expanding, which causes many stars to be farther and farther away from earth, and therefore causes them to be dimmer. But the expansion of the universe only affects the distance between galaxy groups. It does not affect the distance between stars inside a galaxy, or even the distance between galaxies in a group. On galaxy group scales and smaller, local gravity overpowers the universe's expansion. The stars in our Milky Way galaxy and in nearby galaxies are not increasing in their distance from the earth, despite the expansion of the universe. As a result, the stars in our galaxy and in nearby galaxies are not growing dimmer over time. Interestingly, almost all of the stars that you can see in the night sky with your naked eye are in our galaxy. This means that the expansion of the universe will have essentially no effect on the appearance of the night sky to the naked eye, no matter how long we wait. The night sky will not go completely dark because of the expansion of the universe.
However, powerful telescopes can see other galaxies outside of our group. If we wait long enough, the expansion of the universe will cause there to be fewer galaxies for powerful telescopes to look at. As a galaxy's distance from earth significantly increases, its light is spread over a greater area before reaching us, and is therefore dimmer. As time goes on, more and more galaxies will become simply too far away for powerful telescopes to see. Increasing the sensitivity of our telescopes can help, but it can only do so much. The light from receding galaxies is not only dimmer, it is also redshifted. This means that all the different colors in the light are Doppler shifted to lower frequencies because of the receding motion of the galaxy. As time goes on, the speed at which a certain galaxy outside our group is receding increases because of the universe's expansion, and therefore its light gets more and more redshifted. Light that used to be green or red ends up as radio waves. Eventually, the redshift gets so extreme that the light is effectively redshifted down to nothing. No amount of technological progress will enable telescopes to see light that is not there. As a result, in the very distant future, the universe will have expanded so much that the light from all of the stars and galaxies outside of our galaxy group will never reach earth. Astronomers in the distance future will have to be content with studying only our local group of galaxies. But, again, the night sky will still look the same to the naked eye.
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