Show that every positive number a has exactly one positive square root, as follows:(a) Sho...

Show that every positive number a has exactly one positive square root, as follows:

(a) Show that if x > 0 and 0 ≤ h<1, then

(b) Let x > 0. Show that if x2<a, then (x + h)2<a for some h > 0; and if x2 > a, then (x − h)2 > a for some h > 0.

(c) Given a > 0, let B be the set of all real numbers x such that x2<a. Show that B is bounded above and contains at least one positive number.

Let b = sup B; show that b2 = a.

(d) Show that if b and c are positive and b2 = c2, then b = c.