Math 205a Quiz 05 Worksheet - Bates College, 2008 Page 2

Math 205A

Quiz 05 page 2

October 24, 2008

NAME

3b. For each of the 3 properties in problem (1), decide if the property holds for H or does not. If it

holds, explain why; if it does not, then give a counterexample (in pictures if you wish).

(1)

(2)

(3)

4. Let S be the “script-S” vector space of all inﬁnite sequences of real numbers we’ve discussed in class.

Let H be the subset of S consisting of all sequences which change sign from term-to-term, that is, if s is a

sequence in H, and some term of s is positive, the one after it is negative, and vice-versa. By default, we

will also automatically add the sequence 0 = (0, 0, 0, . . . ) to H.

4a. Is s

= (1,

2, 3,

4, 5,

6, . . . ) in H?

4b. Is s

= (3, 0,

3.1, 0, 3.14, 0,

3.141, . . . ) in H?

For each of the 3 properties in problem (1), decide if the property holds for H or does not. If it holds,

explain why; if it does not, then give a counterexample using speciﬁc sequences.

(1)

(2)

(3)

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