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If x= 0 is rational and y irrational, prove that x + y,x − y,xy,x/y and y/x are all irrational.
Let a,b ∈ R with a < b. Let Ta,b ⊆ Q be the subset defined by Ta,b :=r ∈ Q : a<r<b. Prove that the set Ta,b is infinite.
if a,b ∈ R with a < b, then there exists a rational number r ∈Qsuch that a < r <b.
If x is an arbitrary real number, show that there exists a unique n ∈ Z such that n ≤ x <n+1. (This is called the greatest integer in x and denoted by [x].)
If x > 0, show that there exists n ∈ N such that 1/n < x.
If x is a real number, prove that there are integers p, q ∈ Z such that p < x < q.
Write a formula for the an in each of the following sequences {an} ∞ n=1 given by
(i) 2, 1, 4, 3, 6, 5, 8, 7, . . .
(ii) 1, 3, 6, 10, 15, . . .
(iii) 1, −4, 9, −16, 25, −36, . . . Which ones among these are the subsequence of {n} ∞ n=1?
If x and y are arbitrary real numbers with x < y, prove that there exists at least one irrational number z satisfying x < z < y, and hence infinitely many.
Is the sum or product of two irrational numbers always irrational?
If x is not equal to 0 is rational and y irrational, prove that x + y, x − y, xy, x/y and y/x are all irrational.
