UMC Advanced Calculus Numerical Sequences Inequalities & Number Sets Exam Practice
Question Description
In Questions 1-5 prove your answer. In this test you can usewithout proofs theorems that were proved in the lectures or in thebook, just give a reference.
1.(10 pts) Let S be a subset of R, and u ? R. Two of the followingstatements, if combined, imply that u = inf(S). Which two statements?
A: u ? s for every s ? S.
B: There exists ? > 0 and s ? S so that s < u + ?.
C: For every ? > 0, the number ?u?? is not an upper bound of theset?S={x?R: ?x?S}.
D: There exists ? > 0 so that u + ? is not a lower bound of S.
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and let ? > 0. Suppose that |xn| < M for every n. Select one of thefollowing sets of constants ?1 and ?2 for which the inequalities
2. (10 pts) Let {xn} and {yn} be two sequences,limxn =L1 ?=0, limyn =L2 ?=0,
|xn ?L1|<?1imply |xnyn ? L1L2| < ?.
??A: ?1 = 2M , ?2 = 2|L1|.
??B: ?1 = 2|L1|, ?2 = 2M .
??C: ?1 = 2|L2|, ?2 = 2M .
??D: ?1 = 2M , ?2 = 2|L2|.
and |yn ?L2|<?2
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3. (10 pts) Which of the following statements imply that a sequence{xn} does not have a limit:
A: for every ? > 0 there exist n, m ? N such that |xn ? xm| > ?.
B: there exist a natural number K and ? > 0 such that for every
n > K we have |xn ? xK | > ?.
C: {xn} is an increasing sequence and |xn| > 1000 for every n > 1000.
D: there exists ? > 0 such that for every natural number K thereexist m, n > K with |xm ? xn| > ?.
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4.(10 pts) Which two of the following statements combined implythat limx?3 f(x) = 2?
A: limn?? f (3 + 1 ) = limn?? f (3 ? 1 ) = 2.nn
B: limx?3+ f(x) = 2
C: limx?3+ f(x) and limx?3? f(x) exist
D: For every sequence xn ? 3 with xn ? 3 for every n, one has
lim f(xn) = 2.n??
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5. (10 pts) Which two of the following statements combined implythat the equation f (x) = 1 has a solution in the interval [1, 3]?
A: 0 < f(1) and f(3) < 2.B: f is continuous on [1, 3].C: f(1) < 0 and f(3) > 2.D: f is monotone on [1, 3].
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6. (10 pts) Suppose that lim|xn| = 3, but {xn} does not have asubsequence with limit 3. Prove that lim xn = ?3.
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7. (10 pts). Suppose {xn} and {yn} are bounded sequences and forevery n ? N
xn + yn+1 ? xn+1 + yn,
and
Prove that both sequences {xn} and {yn} converge.
xn + yn ? xn+1 + yn+1.
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8. (10 pts) Prove using the definition that the following limit is equalto ?? :
lim x+1 =??.x?2? x2 ? 4
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9. (10 pts) Prove that if a function f is non-negative and continuouson the interval [1, ?), and limx?? f (x) = 0, then there exists xM ?[1,?) such that f(xM) ? f(x) for every x ? [1,?).
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10. (10 pts) Prove that the function f(x) = x1/3 is uniformly con-tinuous on [1, ?).
Hint: Use x ? y = (x1/3 ? y1/3)(x2/3 + x1/3y1/3 + y2/3).
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