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Prof. Alexandru Suciu
MATH 3150 Real Analysis Spring 2011
Midterm Exam
Instructions: Write your name in the space provided. Calculators are permitted, but no notes
are allowed. Each problem is worth 10 points .
1. Let a; = 0, = 1, a Vf, ..., a = 1 Ee,
(a) Show that the sequence {a,} is strictly increasing.
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ea
(b) Show that the sequence {a,} is bounded above.
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$A =O ¢ 3 Sag
o Assume $ an <3 os
oThey i ty. = tba, are sis9= 3
(c) Show that the sequence {a,} is converging. Give a reason for your answer.
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MATH 3150 Midterm Spring 2011
2. Let {x,} be a sequence in a complete metric (X, d).
(a) Suppose d(zni1,%n) < 1/2", for all n > 1. Show that {r,} converges.
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(b) Suppose d(%n4i,%n) < 1/n, for all n > 1. Show by example that {xn} may not
converge.
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MATH 3150 Midterm Spring 2011
5. Let (X,d) be a metric space, A a subset of X, and x a point in X. We say that:
e The point « is an accumulation point for A if every open set U containing x
contains some point of A other than z.
e The point 2 is a limit point for A if every open set U containing «x contains some
point of A.
(a) Suppose « is a limit point for A, and 2 ¢ A; then show that « is an accumulation
point for A.
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(b) Let A = {1/n | n € N}, viewed as a subset of R.
e What are the limit points of A?
e What are the accumulation points of A?
Does the set of limit points coincide with the set of accumulation points?
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MATH 3150 Midterm Spring 2011
6. Decide whether each of the following series converges or not. In each case, indicate
which test is used, and why that test yields the conclusion you are drawing.
(a)
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ee
Be oliveraeo Ga: ee
av
aires senes Avera es
