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    true false decide which of the following statements are true and which are false.  prove the true statements and give counter examples for the false ones.

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    If   is a convergent sequence and   is a divergent sequence,  then    must diverge. If   and   are convergent sequences such that     for every  n ,  then

    if  ,  then     converges. If  and if there exists a sequence   converging to  a such that   ,  then
    a. Prove:  if     and  ,  then     is eventually less than  b.  in other words,  there exists   such that    whenever . Prove: if     and  ,  then    is eventually greater than  a.  if you prove  a.,  then you may say that the proof of  b.
    let   be a sequence of positive terms such that .  prove the following: if  ,  then   ,  hint:  first show that    is eventually deceasing and, therefore, converges why?    then show that the limit must be  0.  note:  problems 2  and  1.

    If ,  then .  hint:  show that    is eventually increasing but does not converge. Let    be a sequence of positive terms.  prove that the infinite series     either converges or diverges to .  in other words,  assume    for every  n   and let     be the sequence of partial sums   .  prove that either    converges or diverges to . Suppose that    and    are two positive term series such that    for every n.

      let   be an open interval containing  a   and suppose  f   and  g   are two functions defined on . Prove: if  ,  then  f   is bounded on a deleted  neighborhood of .  in other words,  show there exists    and  m such that if    then .

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      Prove: if    and  g   is bounded on a deleted  neighborhood of  a ,  then .   do not assume   exists.

      find a continuous function  that is a surjection  onto   but does not have a fixed point.  i.e.    for every 
      let   be a  continuous function such that    and .  prove that  f   has an absolute minimum value on  ℝ.  in other words,  show there exists  ℝ  such that    for every  ℝ.
    from november 14 through november 19, 1995 and from december 16, 1995 to january 6, 1996 the u.s.