Problem Set - Convergence theorems for sequences

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Convergence theorems for sequences

	Prove that a real sequence can have at most one limit.
	Prove that every convergent sequence is Cauchy.
	Prove that every Cauchy sequence which has a convergent subsequence is itself convergent.
	Prove that every Cauchy sequence is bounded.
	Prove that every convergent sequence is bounded.
	Prove that a contractive sequence is Cauchy.
	Prove that a contractive sequence is convergent.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)