MAT320 Introduction to Analysis
Midterm Examination 1

October 10, 1996

WORK ALL PROBLEMS; EACH IS WORTH 25 POINTS.

NOTE: ``Prove'' means give a careful, well-explained proof.

• 1. Let be a sequence of real numbers, and suppose that the two numbers L and L' satisfy

  

  Prove from the definition of limit that L = L'.

• 2. a) Give an example of a sequence which has no convergent subsequences.
  b) Give an example of a sequence which does not have a limit, but which has a subsequence which does have a limit.
  c) Give an example of a bounded set of real numbers which does not contain its least upper bound.

• 3. The Tetrabonacci numbers are defined by , and

  

  for . Suppose that the limit exists and is equal to L. Calculate L.

• 4. Prove that is not rational.

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