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MATH. 439 Take-home   Spring, 2013

Work as many of the following problems as you can.  Be sure your proofs are readable and, for the most part,  rigorous.  You are encouraged to turn in any positive work on a problem,  even if you do not complete all parts.  The exam is due by Friday afternoon,  May  10.

1. (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.
2. If   is a convergent sequence and   is a divergent sequence,  then    must diverge.
3. If   and   are both divergent sequences,  then    must diverge.
4. If   and   are convergent sequences such that     for every  n,  then

1. If  ,  then     converges.
2. If   converges and  ,  then    converges.
3. If   converges to  ,  then  .
4. If  and if there exists a sequence   converging to  a such that   ,  then

.

1. a. Prove:  If     and  ,  then     is eventually less than  b.  (In other words,  there exists   such that    whenever  .)
2. Prove: If     and  ,  then    is eventually greater than  a.  (If you prove  a.,  then you may say that the proof of  b.  is similar.)

1. Let   be a sequence of positive terms such that  .  Prove the following:
2. 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. f.  above may be useful.)
3. If ,  then  .  (Hint:  Show that    is eventually increasing but does not converge.)
4. Find a sequence   such that    and  .

Find a second sequence    such that    and  .

1. a. 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  .)
2. Suppose that    and    are two positive term series such that    for every n.

Prove:  i.  If      diverges,  then   diverges too.

1. If    converges,  then    converges too.

This is known as the comparison test for positive term series.

1. Let   be an open interval containing  a  and suppose  f  and  g  are two functions defined on  .
2. Prove: If  ,  then  f  is bounded on a deleted  -neighborhood of  .  (In other words,  show there exists    and  M such that if    then  .)
3. Prove: If  f  is continuous at  a, then  f  is bounded on  -neighborhood of .  (This is almost a direct consequence of part  a.)
4. Prove: If    and  g  is bounded on a deleted  -neighborhood of  a,  then  .   (Do not assume   exists.)
5. Show by counter example that c. is no longer true if  .

1. Find a continuous function  that is a surjection  (onto)  but does not have a fixed point.  (i.e.    for every  )

1. 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 .)

English

From November 14 through November 19, 1995 and from December 16, 1995 to January 6, 1996 the U.S. government had shut down due to budgetary impasses between the White House and Congress. The shutdown occurred due to dispute between Democratic President Bill Clinton and Republican Speaker of the House Newt Gingrich on the issue of domestic spending cut in the Fiscal year 1996 budget and resulted in bipartisan agreement to balance the budget in seven years’ time. After occupying, the speakership of the House after the 1994-midterm election Newt Gingrich asked to implement policies related to the Republican Party’s 1994 Contract with America. Contract was a campaign document that promised to slash funding for Congressional committee staff and introduction of balance budget amendment to the constitution. Most of the proposal failed to pass the senate and Republican member started their concern to limit the ability of President Clinton to govern.

In response to this move Clinton’s government refused to accept the demand to cut the allocation of Medicare, Medicaid and other non-defence spending for the fiscal year 1996 budget. Gingrich warned the administration to prevent vote on increasing the federal government debt ceiling, which resulted the US in position of defaulting on its outstanding debts. The continuing resolution bill allowed the government to keep holding their work and offices. The Republican congressional representative and Clinton White House failed to reach an amicable agreement on the budget. This results in closing of all non-essential government spending after the expiration of continuing resolution on November 13, 1995.

The Shutdown lasted until November 19 when the congress and president became ready to balance the budget in seven years. The basic ideas were to continue the government services but the main debate over how the balance will be made is not discussed in detail. On December 15 when the second resolution of continuation expire government shut down once again. On the next 22 days, the government and congress struggled to reach an agreement to stop the shutdown but not able to finalize the terms of agreement. Finally, on January 1996 the President and Congress agreed to a seven year balanced budget plan that included modest spending cuts and tax increases.