Point breakdown : Question 1 : a) 3 b) 3 c) 2 d) 3 Question 2 : a) 4 b) 3 c) 3 Question 3 : a) 3 b) 3 c) 4 Question 4 : a) 3 b) 4 c) 3 Question 5 : a) 3 b) 3 c) 3 d) 1 Question 6 : 10 Question 7 : a) 3 b) 3 c) 4 Question 8 : 10 (free points) Question 9 : 10 Question 10 a) 4 b) 6 =========== HOMEWORK REMARKS==== * Please don't hand in your scrap paper, show that you read through your own solution once before you ask me to do it. * Students are copying literally word for word (sometimes just changing names of elements, and failing to change all the names) parts from Rudin (for question 1) or Stackexchange solutions (for question 10). * People should know better than trying everything by contradiction. Example of student proving a complete subset of a complete space is closed : - Prove that S is closed by proving X\S is open - Assume by contradiction X\S is not open - Proves that S is closed by showing that it contains limit points (and failed to do so) - So X\S is open, which is a contradiction - So X\S is open - So S is closed * "arctan is bijective, its inverse is arccot" ... * Question 1. Many students proving that compact => sequentially compact, then prove that Cauchy sequences with a converging subsequence converge (sometimes several times in the same homework). * Question 3 : "x_n=n does not converge because it goes to infinity" what does it mean for d2 ? this is a bounded sequence in (R,d2) * Question 4a) no need to prove | ||x||-||y|| | <= ||x-y|| ... *********** Question 4b) REALLY BIG PROBLEM FOR EVERYONE : Students assuming that open balls are open sets in question 2 "open sets in metric spaces are all unions of open balls and union/finite intersection of open sets are open, then it's true for open balls", this is something I read, which is very circular. Reformulation : Cannot use that open balls are "open", e.g. "the intersection of open balls is open, so I can cover it with open balls" they are using that the balls are a topology to prove that the balls are a topology .... This mistake is concerning MORE THAN 80% of the people. * Question 4b) After proving the closed ball is the closure of the ball, people proceed to showing that it is indeed closed, which repeats their previous arguments * Question 4b) "The empty set is obviously/clearly/vacuously/trivially open", say why, those words don't help. * Question 4c) No need to show that equiv norm have the same converging sequences after showing that they have the same topology ... * Question 6 : Mostly everyone just copied some proof somewhere, some tried to change some notations around but mostly forgot to change it everywhere ... * Question 6 : No need to REPROVE that the euclidean norm is a norm ! * Question 7c) people forgot to prove || || is a norm. * Question 8 : No one did p<=0 so I did not remove points, if we want to enforce it, everyone should get points deducted from that * Question 9 : Often too handwavy, chaining inequalities "off a null set", not keeping track that this set depends on n,m when writing the Cauchy condition. *Question 10b) Many copied a solution from StackExchange that introduces a useless "q