In Exercise 49 of Section 2.2 you are asked to prove that lim vr = vc for all non-negative real numbers c. The following is a partial proof that lim VT = vc for all real numbers c. Complete the proof. Proof: There are three cases to consider: c > 0, c=0 and c 0. Using the elementary identity 43 - 13 = (a - b)(a3 + ab + b]) it follows that and so IVI - Vol = r - C I(Vr)' + VIVC+ ( VC)] (x3/3 + 1/3cl/3+ c2/3 (1) provided that p2/3 + p1/31/3 + (3/3 2 0. To prove that lim VI = vc it must be shown that Definition 2.2.1 holds, so let e > 0. It I C must be shown that there is some $ > 0 such that | VI-VC) 0 be so small that & 0 and |r - c| c/2. Hence (c/2)3/3 + (c/2)1/31/3 + 2/3 = 12/3((1/2)2/3 + (1/2)1/3 + 1) > 0 (3) Combining Equation 1 and Inequality 3 and using that |r - c)