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practiceexam2

practiceexam2 - n ∞ a n = 0(b Show that Σ n n 2 1...

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MATH 131A - Practice Midterm 2. Please write clearly, and show your reasoning with mathematical rigor. You may use any correct rule about the algebra or order structure of R from Section 3 without proving it. 1. Suppose ( s n ) is a given sequence of nonnegative real numbers, and suppose limsup s n 1. Show that there is a convergent subsequence. (Hint: First show that s n is bounded for n suﬃciently large.) 2. (a) Show that if Σ a n converges, then lim
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Unformatted text preview: n ∞ a n = 0. (b) Show that Σ n n 2 +1 diverges. 3. (a) State the deﬁnition of continuity for a function f at a point x ∈ dom ( f ). (b) Show that if f is continuous at (0 , 1) and if lim x → + f ( x ) exists, then there exists a continuous extension ˜ f of f on [0 , 1) . 4. Give an example of function f : [0 , 1] → [0 , 1] which is discontinuous at every rational points in [0 , 1]. Explain why. 1...
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