## Exam

The exam is posted online on Jan 30, 2017, and must be submitted on March 16, 2018 (the first day of the new semester).

Its goals are, besides testing your newly acquired skills in the Analysis, to teach you a few extra things and check your ability for logical reasoning, not your proficiency in performing long computations.

If you find yourself mired in heavy computations, double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers. They really may change the meaning of what you write!

Problems are subdivided into items. The order of these items is by far non-random, you have to solve them from the first through the last, (solution of one item may be a building block for the next one). A complete solution of a problem is a proof of some important theorem in Analysis, so you will be discovering these results on your own. The Remarks will help you to place the freshly proved theorem on the general picture.

**To get the maximal grade 100, it is not necessary to solve all problems**. Problems are of varying length, variable complexity, various level of abstraction. No apriori points are assigned for solution of each problem, no summation at the end. You can get extra points for short and elegant argument or have some points removed for writing an obviously stupid things (honest errors will simply bring you zero points). You have all the time, try to solve as many problems as you can, we will appreciate and assess the results as objectively and honestly as possible.

You are assumed to work individually, which is, of course, impossible to verify, but please in any case avoid submitting isomorphic solutions: this is a bad taste for take-home exams.

You are absolutely free to write in English (easier for me) or Hebrew, submit handwritten pages or compuscripts, in a hard copy or by email (even scans will work). I we will encounter difficulties reading your submission, we’ll let you know.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (*errare humanum est*), the problem will be either edited (in case of minor omissions) or cancelled (on my account). In case of any doubt don’t hesitate to leave your questions as talkbacks to this post, so that other people will be able to follow. Asking questions is never penalized!