MT2002 Analysis

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Exercises 4

  1. Prove the Triangle inequality:
    If a, b are real numbers then | a + b | lte | a | + | b |.
    When is this inequality strict ?
    Why is it called the Triangle inequality ?
    Use induction to prove that if a1, a2, ..., anare any real numbers then |sigmai1nai| lte sigmai1n|ai|.

    Solution to question 1

  2. Prove the Squeeze rule:
    If anlte xnlte bn for all n and the sequences (an) and (bn) converge to the same limit, then (xn) also converges to this limit.

    Solution to question 2

  3. Give examples or prove the non-existence of sequences which are:
    (a) convergent but not monotonic,
    (b) bounded but not convergent,
    (c) convergent but not bounded,
    (d) monotonic but not bounded,
    (e) divergent to +infinity but not monotonic,
    (f) monotonic and bounded but not convergent,
    (g) unbounded but not monotonic,
    (h) positive and convergent, but not monotonic.

    Solution to question 3

  4. The sequence ( (2n + 1)/(n + 1) ) converges to 2. How big must we choose N to be so that the terms of the sequence are within epsilon of this limit if epsilon = 0.1, 0.01, 0.0001 ?

    Do the same thing for the sequence ( (2n2+ 1)/(n2+1) ).

    Solution to question 4

  5. Give examples or prove the non-existence of sequences such that:
    (a) (|an|) converges but (an) does not converge,
    (b) (an) converges but (|an|) does not converge,
    (c) (an) and (bn) do not converge, but (an+ bn) converges,
    (d) (an) and (an+ bn) converge but (bn) does not converge.

    Solution to question 5

  6. Let a be a real number in (0, 1). Let an be the real number obtained by deleting the first n digits of the decimal expansion of a.
    For example, if a = 0.1234567... then a1 = 0.234567.., a2 = 0.34567..., etc.
    For which a is the sequence (an) convergent and what is its limit ?

    Solution to question 6

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JOC September 2002