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midterm - ∈ R sup f x c x ∈ A = c sup f x x ∈ A...

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Mathematics 131A/2 Fall 2007 Midterm Your Name (Last, First, Middle): Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Point Points Problem Value Received 1 10 2 10 3 10 4 10 5 10 Total 50
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Problem 1 (10 points) Print your name: Let { a n } and { b n } be two convergent sequences of real numbers. Show that the sequence { max ( a n , b n ) } is also convergent.
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Problem 2 (10 points) Print your name: Let A be a nonempty set and let f : A R be a function such that the range f ( A ) is bounded above. Show that for each c R , sup { f ( x ) +
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Unformatted text preview: ∈ R , sup { f ( x ) + c ; x ∈ A } = c + sup { f ( x ); x ∈ A } . Problem 3 (10 points) Print your name: Show that the sequence a n = (-1) n p 2 + 1 n 2 P , n = 1 , 2 , . . . is not a Cauchy sequence. Problem 4 (10 points) Print your name: Let S be a set of sequences of rational numbers such that each sequence has a Fnite number of nonzero terms. Prove that S is countable. Problem 5 (10 points) Print your name: Let { a n } be a monotone sequence of real numbers which has a convergent subsequence. Show that { a n } converges....
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