Exercises

Before giving some comments about the first exercise, here are some general comments about submitting exercises (the comments are sorted by some order): Read the exercise at least once before you submit it. Check that what is written is what you intended to write. Check that it is also correct. If you use some claim, state it exactly, and give some reference to it (e.g., proved in class). If the question has several parts, try to understand how do the parts combine. See if you can make the argument simpler (perhaps using ideas discussed in class). Try to understand the purpose of the question (or the person who gave the question).

Going back to the first exercise, here are comments about the solutions you submitted
(comment 1. corresponds to question 1., for example).

1. To calculate a limit, one needs to show that the limit actually exists, and so one needs to consider every n (not just even n's, for example).
2. This question is all about doing some exact calculation. So writing ~ without explaining what do they mean, and using non-integer m(t) are missing answers.
3. This question has the largest amount of solutions: some of the answers are short and precise, some of the answers are correct and long, and some of the answers can actually be made into correct argumets, but lack the needed definitions and/or details.
4. (a) There is a subtle issue, which some of you addressed, that the statement holds with probability 1 (and not for every value that variables might take). (b) After the comments above, some of you did not use (a) - this was needed.

Regarding the grades for the exercises, we decided to give a grade from 0 to 10 for each question. The final grade will somehow depend on these numbers.