Real Analysis Answers

Let a,b є R, and suppose that for every ε > 0 we have a ≤ b+ε. show that a ≤ b.

Show that if& a < b, then a< (1/2)(a+b) < b&

Show that if a > 0, then (1/a) > 0 and (1/(1/a)) = a.

Show that there does not exist a rational number t such t[sup]2[/sup] = 3.

If a ≠ 0 and b ≠ 0, show that (1/ab) = (1/a)(1/b).

If 0 < a < b, show that
1)& a < (ab)[sup]1/2[/sup] < b;
2) 1/b < 1/a.

If a,b ϵ R, show that a[sup]2[/sup] + b[sup]2[/sup] = 0, if and only if a = 0 and b = 0

show that;
- (- 1 ) a = a

Show that for any number A, the series Summation of (A^n)/n! converges absolutely, conclude that lim as n goes to infinity (A^n)/n! =0

Let f:ℝ￫ℝ be continuous on ℝ, and let P:={ x∈ℝ : f(x) > 0 }. If c∈P, show that there exists a neighborhood V_δ (c) ⊆ P