Test whether the follwing sets are compact or NOT?

I am trying to find which of the following sets are compact and possibly a valid reason for why so I can better understand the concept.

Here are the following sets I am trying to find:

1) ${(x,y) in mathbb{ R}^2:2x^2-y^2 leqslant 1}$.

2) ${x in mathbb{R}^n: 2leqslant ||x||leqslant 4}$.

3) ${(e^{-x} cos x,e^{-x} sin x): x geqslant 0} cup {(x,0): 0leqslant xleqslant 1}$.

Answer

A set $E$ is bounded if it is contained in a ball of finite radius.

1) is not bounded, consider $(0,n) forall ninmathbb{N}$.

2) is closed and bounded in $mathbb{R}^n$, so it’s compact.

3) is also closed and bounded, hence compact, see the reason here.

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