Real Analysis Answers

1) in the system of real numbers the axiom of existence of additive inverse states that, for all x element of R there exists y element of R such that x+y=y+x=0. prove that the additive inverse(y) corresponding to each real number x is unique. what can you say about the statement, there exists y element of R such that for all x element of R x+y=y+x=0?

2)if a and b are irrational numbers is, a to the power of b necessarily an irrational number? prove your claim

3)suppose A,B,C,D are four distinct points with position vectors a,b,c,d respectively, show that A,B,C,D lie on a plane if and only if there exists w,x,y,z element of R such that w+x+y+z=0 and aw+bx+cy+dz=0

if A be a subset of real number and B is real number then show that sup(b+a)=b+sup(A)

Prove that
lim┬(x□(→┬ )1^- )⁡〖f(x)≔〗 lim┬(x□(→┬ )1^- )⁡〖 (x+2)/(2x^2-3x+1)=-∞.〗

Let E⊆R be nonempty. Prove that:

(i)E has an infimum if and only if –E has a supremum, in which case

sup⁡(-E)=-inf⁡E.

Show that there are at least three distinct points x1,x2, and x3 such that f(x1)=f(x2)=f(x3)=10, where f(x)=x^3/(x^2-1)

I was wondering if you would be able to give me a solution or hint on the following problem:

Let D be a non-empty subset of the real numbers, and E={ax, x in D} where a>0. Prove that E is open if and only if D is open.

Any help would be greatly appreciated. Thank you

the set of _____ is the set {...,-5,-4,-3,-2,-1,0,1,2,3,2,4,5,...}

Let A={1,1/2,1/4,1/8,…} and B={ 1/2,3/4,7/8,…}. Explain why supA=supB=1?

What is Bolzano-Weierstrass Theorem? Give the proof.

Show that f(x)=x^2 is not uniformly continuous on R.