Real numbers Definition and 214 Threads

i know that the set "all real numbers" make up a vector space, but how do you prove that it is so?

how do you prove the set of vectors "all ordered quadruples of positive real numbers" make a vector space?

Hi people, I need some help with these questions please: 1.Is the set of all x in the real numbers such that (x+pi) is rational, countable? I don't think this is countable, isn't the only possible value for x = -pi, all other irrationals will not make x+pi rational i thought? 2.Is...

Let S be a finite set of real numbers. What is a natural inner product to define on the space of all functions f:S->R? I want to approximate an arbitrary function with a polynomial of a fixed degree (both of which are defined only on S), and I want to use projections to do it, but I have no...

Hello, I'm working on this problem: If t(P,Q)=max(|x1 - x2|,|y1 - y2|), show that t is a metric for the set of all ordered pairs of real numbers. I have proved the first three parts of the definition of a metric 1) t(P,Q) >0 2) t(P,Q) =0 IFF P=Q 3) t(P,Q) = t(Q,P) all not so hard. I'm...

So the problem, and my partial solution are in the attached PDF. I would like feedback on my proof of the first statement, if it is technically correct and if it is good. Any ideas as to how I can use/generalize/extend the present proof to proof the second statement, namely that E (the Cantor...

This is something that I think I should already know, but I am confused. It really seems to me that the set of all real numbers, \Re should be compact. However, this would require that \Re be closed and bounded, or equivalently, that every sequence of points in \Re have a limit...

prove that between any two real numbers there is a number of the form \frac{k}{2^n} where k is an integer and n is a natural number.

I have three problems that I can't seem to solve. I was wondering if anybody could help me or explain to me how to solve these. Note: * = multiplication. 1. (6^1/2 * 2^1/3)^6 2. ^4√7 + 2^4√1792 3. 3(X-4)^1/2 + 5 = 11

Why does commutivity of real numbers exist for multiplication ie why a*b=b*a ? Roger

1.Does a sequence exist that has every point of R(real numbers) as an accumulation point? 2.Show that closed is essential in the Heine-Borel Theorem by finding an open cover of a non-closed bounded set that does not have a finite sub-cover. I think that the set of rational numbers has...

Ok...here is some back ground into my new found situation. I have done very well in every math class up to this point in time so I felt it was time for me to start looking at taking some more difficult classes. That being said I am technically still in my freshman year in college so I may have...

Ok, I'm sure this is an easy problem and all, but it's pissing me off. I'm probably just not understanding it. The function f has the property that for any real numbers x and y, f(x+y)=f(x)+f(y)+1. If f(1)=2, what is f(3)? help.

I just learned something really cool. Choose 13 real numbers x_1,x_2,\ldots,x_{13}\in\mathbbb{R} with x_i\neq x_j if i\neq j. For these 13 numbers there exist at least two numbers amongst them such that 0 \; < \; \frac{x_i-x_j}{1+x_ix_j} \; \leq \; 2-\sqrt{3} Isn't that cool?! (I...