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I found a theorem that the $S^2$ is connected however I cannot find a proof via Google. Is there any hint how to proof that the $S^2$ sphere is connected?

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    Also: do you know what arc-connectedness is and that it implies connectedness? otherwise, all of the answers so far won't help much... This is precisely why it is good to explain your background!2011-07-25

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Probably the easiest way is to notice that $S^2$ path-connected, which implies connectedness. You can join any two points on $S^2$ with a segment of a great circle.

Or you can notice that $\mathbb{R}^3-\{0\}$ is path connected, and that there exists a continuous surjective map $\mathbb{R}^3-\{0\} \to S^2$.