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I marked a place, which confuses me:

So, how do we know, that if A is complete it means, that sequence ${x_n}$ converges to a point, that belongs to A?

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    Isn't that just the definition of complete? Note that $x_n$ is a Cauchy-sequence, not any sequence. – 2017-02-08
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    Yes, you are right. I just have written the definiton of comlete incorrectly. Sorry. – 2017-02-08
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    It is just the definition of complete subset. – 2017-02-08

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