First we will indicate a couple of notable gaps in the proof.
First, there is a gap between the sentences: Assume for
the sake of contradiction that S has no
PROBLEM SET #6 , 18.100C
Problem 1. If f is a continuous mapping of a metric space X into a metric space
Y , prove that f (E) f (E). Show by an example that f (E) can be a proper subset
of f (E).
Solution. If f (E) is empty then the pr
24. Let X be a metric space in which every infinite subset has a
limit point. Prove that X is separable. Hint: Fix > 0, and pick x1
X. Having chosen x1,.,xj X, choose xj+1, if possible, so that
d(xi, xj+1) for i = 1, ., j. Show that this process must sto
PEER REVIEW, 18.100C
FOR CASEY FU
This is review of the paper will consist of two parts. Th rst part is a friendly
review and suggestions on the logic and formatting of the paper. The second part
suggests some more spec
PAPER REVIEW, 18.100C
FOR KEVIN Y. CHEN
This is review of the paper will consist of two parts. The rst part includes
friendly notes and comments on the mathematical context and logic of the paper.
The second part focuse
PEER REVIEW, 18.100C
FOR A. SCHVARTZMAN
First of all I have to say that you did a great job and your communication on
each audience is praise worthy! It was a great pleasure to read your paper! Way
to go! They were hard
WRITING ASSIGNMENT #3 , 18.100C
In this paper we shall dene and prove wherever applicable, the triangle inequality to three dierent audiences. We specically chose Michael Artin, Arnold
Schwarzenegger and Pythagoras.
18.100C: A DEEPER LOOK INTO REAL SEGMENTS
A BSTRACT. In this paper we encounter the question whether a subset of an uncountable
set can be countable under certain properties. Further questioning and extensions are provided.
PROBLEM SET #9 , 18.100C
Problem 1. Recall that B 1 ([a, b]) is dened as the space of functions f : [a, b] R
which are everywhere dierentiable and whose derivative f ? is a bounded function.
One equips this space with the metric
PROBLEM SET #7 , 18.100C
Problem 1. Recall that in class, we have dened cos(x) = 1 x2 /2 + x4 /4! .
Suppose that we dene a number /2 as follows: it is the smallest positive number
such that cos(/2) = 0. Show that this denition makes se