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Math Help - Sequences, series, completeness

  1. #1
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    Sequences, series, completeness

    Some questions that I can't quite get my head around:


    1) Show that every sequence which is not bounded above has an increasing subsequence.

    2) Show that if (an) \rightarrow 0 then there is a subsequence of (\frac {1} {an}) which tends to + \infty , or there is a subsequence of (\frac {1} {an}) which tends to - \infty .

    3) If a is irrational and b is not equal to 0 and b is an integer, is  \frac {a} {b} rational or irrational? Justify your answer.

    irrational. justification?

    4) What is the value of  \lim_{n \to \infty} \ (3^n + 5^n+ 2^n)^{\frac {1} {n}} ?

    5) State, without proof, whether or not there can exist a Cauchy sequences (a_n) in R (real numbers), which does not converge in R (real numbers).

    No seems very obvious.
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  2. #2
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    If \{ a_n \} is unbounded above, take  b_1= a_1 and find b_n recursively: Given \{ b_n \}_{n \leq m} there exists b_m such that b_m \geq b_n for all n \leq m, because otherwise taking M= max_{n \leq m}{\vert b_n \vert } would be a bound of \{a_n \}.

    If \{a_n \} \longrightarrow 0 then \{ \frac{1}{a_n} \} is unbounded, and by the previous exercise has an increasing or decrasing subsequence.

    If a is irrational, b an integer and q= \frac{a}{b} were a rational number, since \mathbb{Q} is a field then qb would be a rational number but qb=a. therefore q is irrational.

    The answer to the limit is five, although I use the fact that you want the limit of the n-norm in Euclidean 3-space of the vector (3,5,2) which is the \infty-norm of (3,5,2) which is max \{2,3,5 \}=5. If you've seen this I'll elaborate, if not then use this only as a guide to the answer.

    And finally, No, there cannot exist Cauchy sequences which do not converge because \mathbb{R} is complete
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