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Well, seeing several galaxies move further and further away is one thing, but concluding the whole universe is expanding is another step. I'm especially curious about the details why and how they exactly came to that conclusion.

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As far as I know this is the only reason why they come to that conclusion

At the beginning , Einstein did not beleive in an expanding universe but Hubbles succeeded in convincing him

The theory of the relativity itself admit any kind of universe : expanding - stable - contracting