Question: (a) Prove that every convergent sequence is bounded. That is, if (pn) converges in the metric spaceM, prove that there is some neighborhoodMrq containing the

  1. (a) Prove that every convergent sequence is bounded. That is, if (pn) converges in the metric spaceM, prove that there is some neighborhoodMrq containing the set{pn:nN}.

(b) Is the same true for a Cauchy sequence in an incomplete metric space?

2.A mapf:MNisopenif for each open setUM, the image setf(U) is open inN.

(c)Iffis an open, continuous bijection, is it a homeomorphism?

(d)Iff:RRis a continuous surjection, must it be open?

(e)Iff:RRis a continuous, open surjection, must it be a homeomorphism?

(f)What happens in (e) ifRis replaced by the unit circleS1?

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