MA 425 content specifications
The content specifications are:
A  B  C  
sequential characterization
(2:1.4.2, 3:4, 5:2, 5:5, 5:3.4.6) 
formulate and prove  prove at least 2  prove at least 1 
Cauchy criteria  formulate and prove  prove at least 1
(5:3.2.5, 4) 
use 1 
Archimedean Principles  prove all are equivalent
(2:6 and 2:8 together) 
use each of AP1, AP2, AP3  
forms of completeness  use Nested Intervals  prove 2 are equivalent
(2:5,4:2, 4:4) 

compactness  use opencovers version  use HeineBorel version  use sequential compactness 
limits and continuity  use topological version of continuity to show a function is continuous 

use the sequential criterion for continuity to show a function is continuous 
major/named theorems  prove 2 ``at infinity" versions of named theorems
(7:8, 8:3, 9: 16) 
prove an ``at infinity" version of a named theorem
(7:8, 8: 3, 9: 16) 
prove at least one
(6:11, 7:5) 
derivatives  use Taylor's Theorem 

use the limit definition of the derivative 
integration  compute 3 integrals with proof  compute 2 integrals with proof  compute 1 integral with proof 
convergence of functions  prove a sequence converges pointwise but not uniformly  
series  prove series versions of 2 named theorems for sequences  prove series version of a named theorem for sequences  
explanation points  3 points  2 points  1 point 
pathologies & exotica points  3 points  2 points  1 point 
topology points  3 points  2 points  1 point 
 Numbers in italics represent problems that definitely satisfy that specification. 3:4 means problem 4 from homework 3. 2:1.4.8 means Abbott's exercise 8 from chapter 1.4, which was assigned on homework 2.
 formulate and prove means you must come up with the statement of the theorem, in addition to proving it.
 use means to use the result in question in the course of a proof
 a named theorem is one with a person's name attached (such as the BolzanoWeierstrass Theorem) or a phrase (such as the Mean Value Theorem)
 a sequential characterization or a sequential criterion is a theorem of the form [statement not involving sequences] if and only if [statement involving sequences]. Examples include:
 The sequential characterization of supremum.
 The sequential criterion for continuity.
 The sequential characterization of closed sets.
 forms of completeness are: the Monotone Sequence Property, the Least Upper Bound Property, the Greatest Lower Bound Property, and the BolzanoWeierstrass Property
Points can be earned by completing the following problems. Topology points may also be earned by using arguments that reference open, closed, and/or compact sets. Explanation points may also be earned by giving a very clear unsolicited explanation as part of one of your proofs.
explanation  2:2, 3:2, 8:4, 10:9 
pathologies & exotica  1:3,
2: 1.4.8, 6: 6, 3.3.7 9: 7b 10: 7.3.9(d), 8 
topology  7:2
10:10 