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
- (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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Financial Algebra advanced algebra with financial applications
Authors: Robert K. Gerver
1st edition
978-1285444857, 128544485X, 978-0357229101, 035722910X, 978-0538449670
