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This is a problem I have been trying to understand but i simply cannot.

I understand that the length of y must be greater than 0 but I do not understand how to show B_2 is a regular language or why pumping up does not work in part ii.

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    $\begingroup$ What's $\Sigma$? Tried anything to attack either question? As for why this language is regular, see whether $000010$ is, or $0011000000000$ $\endgroup$ – G. Bach Feb 21 '16 at 22:25
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For (i), the pumping lemma says there is some such division, not that any division you happen to choose has to work. For (ii), because any string starting and ending in $0$ is in the language ($x$ is anything whatsoever in it's definition).

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