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1. Can someone explain why this set is unbound closed set( -infinity,0 ] obviously, the upper...

1. Can someone explain why this set is unbound closed set( -infinity,0 ] obviously, the upper bound is 0 because of ] but why it is actually unbounded set? Please follow the comment.

2. why open subset can lead us to convergence. I don't understand it. Please give me some example or other theorem to support it

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