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