kavlas

No. By ~[math]A\in A[/math] i mean ~[math](A\in A)[/math]

If you consider ~[math]A\in A[/math] as an axiom then how would you prove whether [math]A\in B[/math] and [math]B\in A[/math] is true or false?

I did not ask why ,but how can statements along a proof be demonstrated to be true. Any way thanks for the help so far. I did a google journey but it was not very satisfactory. Everything is so obscure and not very clear w.r.t the mechanisms of a proof. I wander is it so difficult to really analyse a mathematical proof?? I also wander what are the constituents of a mathematical proof

And how can statements be demonstrated true??

yes but you do not show how: [math]-M\leq -|M_{2}|[/math]? Also how do you know that:[math]M_{2} \leq -|a|[/math] ??

How do you define logical consequence?

If we accept that the the axiom of regularity doe not allow that,how do we then prove that. I mean how do we prove that: ~[math]A\in A[/math]

I DID not define A = {x : ~xεx } SO i am not asking for the Russel's Paradox. I am simply asking if we can prove that [math] A \in A [/math] is true or false

how do we prove that : AεΑ is true or false ??

But in real Nos zero is not defined as you mention in your second line of proof. How do you define "-" in real Nos. I know that "-" in real Nos is defined by the equation : x+(-y) = x-y

This is the real zero

prove : -0=0

To find the above limit you need the following theorem: [math]lim_{x\to\infty} f(x)=m\Longrightarrow lim_{x\to\infty} [f(x)]^n = m^n[/math] for all natural Nos n

Apart from the assumption that a=0,i am sorry ,but i cannot see any other assumptions for E. Please ,explain

You mean that the problem ,apart from the proof, is not correct. Because let us suppose that: E ={1/n : nεN},then the 2nd sequence [math]y_{n}=\frac{2}{n}[/math] does not lie in E . So unless we specify E the problem is not provable. The proof is not mine ,it was suggested to me ,as i noted in the OP

I don't know ,man, i tried to find any theorem connecting sequences with uniform continuity and i could not.The only theorem i found was the well known one connecting simple continuity with sequences. And that contradiction part it is not so clear to me. What do you say

Given: 1) E is a subset of real Nos not closed 2) a ,is an accumulation point of E not belonging to E 3) f is a function of E to R (=real Nos) ,where f(x)= 1/(x-a) and ,x ,belongs to E Then prove that f is not uniformly continuous in E The following proof was suggested ,but i am not quite sure about it Proof: Let be ε>0 given. Suppose the function IS uniformly continuous (seeking a contradiction). Then there exists δ>0 such that for all x,y belonging to the real Nos with |x-y|<δ , we get |f(x)-f(y)|<ε . (The last bit being less than ,ε, is what we are going to contradict.) In particular, we can pick ε = 1/2 and then there exists a δ>0 such that the above holds. WLOG, let a=0 . Then take [math] x_{n}=\frac{1}{n}[/math] and [math]y_{n}=\frac{2}{n}[/math] . With this, there certainly exists a natural No k such that for [math]n\geq k[/math], we get [math]|x_{n}-y_{n}|<\delta[/math] . But then notice the following (we are assuming a=0 , so f(x)=1/x ). [math]|f(x_{n})-f(y_{n})| = |\frac{1}{x_{n}}-\frac{1}{y_{n}}|[/math][math] = |n-\frac{n}{2}|=\frac{n}{2}\geq\frac{1}{2}=\epsilon[/math] Contradicting the uniform continuity assumption ,

As i said in my other post ,i forgot to mention it in my OP

you mean that :[math]\delta[/math]=min{[math] 1,\frac{\epsilon}{(a^2+3|a|+2)}[/math]} ??

Sorry ,i did not mentioned it in my original post ,but i meant to solve the problem using the epsilon - delta definition

I was trying to prove that the function: [math]f(x)=\frac{x+1}{x^2+1}[/math] is continuous over the real Nos And in considering |f(x)-f(a)| I come up with the inequality:[math]|f(x)-f(a)|\leq\frac{|x-a|(|ax|+|a|+|x|+1)}{(x^2+1)(a^2+1)}[/math] And in taking values of x near a ,i.e |a-x|<1 i come up with the inequality:[math]\frac{|x-a|(|ax|+|a|+|x|+1)}{(x^2+1)(a^2+1)}\leq\frac{|a-x|(|a|^2+3|a|+2)}{(x^2+1)(a^2+1)}[/math] And here i stop Any ideas how to get rid of x^2+1 in the denominator??

Anyway ,do the charges Q and 2Q produce the same deceleration on the falling body??

You mean that,if a charge Q produces an ,x deceleration,then a 2Q charge will produce the same deceleration??

A sequence [math]x_n[/math] converges iff there exists, a, such that :[math] lim_{n\rightarrow\infty} x_{n}= a[/math]

How do we prove it exists??