05sprg-test2

05sprg-test2 - MAT 371 March 9, 2005 Advanced Calculus Test...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MAT 371 Advanced Calculus /120 March 9, 2005 Test 2 name 1 2 3 4 5 6 20 20 20 20 20 20 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 1. a. Define limit point (of a set) and limit (of a function). b. Define open set and closed set . c. State the ε - δ -definitions of continuous and uniformly continuous 2. a. Without proof, find all limit points for each of the following sets (as subsets of R ). (i) Z . (ii) Q . (ii) [0 , 1). (iv) { ( - 1) n (1+ n ) n : n Z + } . Bonus : { sin n : n Z + } . b. Working from the definition, prove that every constant sequence converges. c. Prove that the sequence ( a n ) n =1 defined by a n = ( - 1) n n +1 n does not converge. 3. Suppose that M R and ( a n ) n =1 is a sequence in R that converges to L R . Prove that if for all n , a n < M , then L M . 4. a. Suppose F X is a closed set in a metric space X .
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online