Mathematics 741 - Advanced Probability Theory I - Fall 2008 - ZirbelAnalysis warmup.These problems are representative of the analysis problems that arise in Mathematics741.They are in no particular order and are of varying difficulty levels, so try them allrather than stopping when you find a problem you cannot solve. Attempt all problems. Iam interested in seeing partial solutions.1.LetX: Ω→R.SetX+= max(0, X) andX-= max(0,-X).Show thatX=X+-X-and that|X|=X++X-. Do this by consideringX(ω) for a givenωin Ω.2.The squeeze law.Suppose thatan≤bn≤cnfor alln= 1,2, . . ., and that (an) and(cn) converge to the same finite limitLasn→ ∞. Show that (bn) converges toLaswell. Use anε-δargument, not lim sup.3.If (an) converges to a finite limitaasn→ ∞, show thatlimN→∞1NNXn=1an=a.4.LetT⊂R, leta= sup(T), and suppose thata <∞and thata /∈T. Show that thereexists a sequence (an)⊂Tfor whichlimn→∞an=a.
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