3. Recall that a rational number r EQ is a number of the form r = where a, b & Z, b +0. Decide whether each of the following claims is true or false. Then prove your answers. Claim 1. There is a positive rational number smaller than any other positive rational number. Claim 2. For any r € Q with r = 0, there exists m € Z for which rm € N = {1,2,3, ...}. 4. Suppose that r € Q, r‡ 0, and that x € R\Q (x is irrational). Prove that rx is also irrational. 5. Prove that for any two nonnegative real numbers, their arithmetic mean is always greater than or equal to their geometric mean. Meaning, prove that for any x,y ER where x, y ≥ 0, x+y 2 Σvry. Moreover, prove that there is equality if and only if x = y.

Let Q be the set of all rational numbers. Which of these equations has no solution in Q? I: x? – 2 = 0 A. I and II only B. II only С. Ш only D. II and III only II: 2x2 +1 = 7 III: x3 – 4 = -31 E. None of the above.

I've understood abc but d. For a, Cn = close interval -1/N ，1/N For b, C =0 For c, Dn = open interval -N, N What is the correct steps for part d? By letting D goes to infinity?

#1 Thanks.

In which pair do both decimals represent rational numbers?     O A. 2.71828182845904523536... and 0.1222 O B. 3.141592653589793238... and 0.684 O C. 3.141592653589793238... and 0.1222.. .O D. 0.684 and 0.1222...