Why don’t we constantly see new stars in the sky as an increase of light travels to us?

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with how light works and the constant expansion of what we term the “observable universe” why don’t we constantly see new stars appearing in the night sky as the observable part expands and stars/galaxies light reaches us for the first time?

The night sky has stayed relatively the same (accounting for changing postions over time, stella phenom, supernovas etc.) for all of humans written history.

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Any new Star light to reach us would be from the new furthest star which would be the new dimmest star in the sky. Even under the best viewing conditions we can only see a very small fraction of stars from earth so these new stars would certainly not be bright enough to be seen.

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