Could someone please check my work

Prove That The Sequence is monotone and bounded. Then find The limit. Sa = and Snap = 4 25. + 1 for ell nerN S, = 2 5 , = 472( 2 ) + 1 = 15 = 0.5 5 1016 9944 Sa # 4 5 2 ( 5 ) + 1 = 0. 36 38 36 6 726 54 - 4 2(0. 36 58 361 726) + 1 = 0. 328602471 2 Ss = 4 2(0. 328603471 2) +1 = 0. 32183/1 807 The sequence, ( Sn), seems to be decreasing and bounded below by 0.3 To prove ( sn) is bounded below by 0.3, apply induction : Base step : For A = 1 , 5 , # 2 > 0. 3 Induction Hypothesis: Suppose $ 70.3 for 3KGIN Induction Step : 5 k, = 4 525 * + 1 > # 12 (0. 3) + 1 =0. 316 2 2776 6 70.3 Thus, S, 30.3 for all neIN, which means ( Sn) is bounded below by 0.3. To prove (s,) is decreasing and Thus Monotone, apply Base step : For nel, S, = 2 and $2 = 0. 5 5 90 16 9 9induction : => S, 7 52. Induction Hypothesis : Suppose SK > SKY, for IKEIN. Induction Step : SK+1 = 4 25( +1 7 4 12ski+ 1 = 5k+ 2 Thus , Sn > Sny, for all nell. Hence, ( s, ) is a decreasing MonoTone Sequence bounded by [0.3, 2] . By The Monotone Convergence Theorem ( Theorem 4.3.3), since ( S. ) is a bounded MonoTone Sequence, Then (S.) is convergent. To find The limit of ( sn) : s= lim Sn = lim sn+ = lim + Jasntl S = 7 lim vasn+ 1 by Theorem 4.2. 1 ( 6 ) S = 4 -lim (25n+1) S = - Vlim ( 25,) + 1 by Theorem 4.2.1 (6) 5 = 4 2lims, + 1 by Theorem 4.2. 1 (6) S = 4 125+ 1 45 = 125+ 1 165 2= 25+ 1 165 3- 25 -1= 0 - -2+ 1(-2)"-4(16)(-1) S = by the quadratic 2 ( 16 ) formula 2 168 32 S = 0. 3201 7 41 016 - 0. 17 5 1 941016 Since 5 >0 => 5= 0. 3201941016 Hence, The limit of ( S. ) is 0. 3201941016