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It really seems to me that the set of all real numbers, [itex]\Re[/itex] should be compact.

However, this would require that [itex]\Re[/itex] be closed and bounded, or equivalently,

that every sequence of points in [itex]\Re[/itex] have a limit point in [itex]\Re[/itex].

But I don't see how [itex]\Re[/itex] can be closed and bounded, unless we somehow decide that infinity is a real number.

So, are the real numbers compact?

thanks