Polynomial Interpolation

Please see attachment!

Polynomial Interpolation

See Attachment!

Characterizing the metric space {N}

For the metric space { N }, the set of all natural numbers, characterize whether or not it has the following properties: compact, totally bounded, has the Heine-Borel property, complete. For compact, we are to show that every sequence converges. For totally bounded, we are to show that it can be covered by finitely many sets of diameter less than epsilon. For Heine-Borel, we are to show there is a finite subcover. And for completeness, we are to show that each Cauchy sequence converges. We are not allowed to use the fact that compactness implies completeness, etc. We can use definitions only.

Characterize the Real numbers with the Discrete Metric

Characterize the set of all real numbers with the discrete metric as to whether it is compact, complete, or totally bounded. Use definitions only! (i.e. compact => every sequence converges, etc)

Show that a subset can be covered by one open ball

If a subset A of a metric space X has diameter less than epsilon, then it can be covered with one open ball of radius epsilon. Prove. (We must use direct definitions only for the proof).