Could someone please check my work

HWLO QL2 Prove That The Sequence is moncrone and bounded. Then find the limit. Sa=aand Snap = 4 25, +1 for all ne IN S, = 2 S, * 4 2( 2)+ 1 = 15 = 0.5 5 1016 9944 S, = 452 (5 ) + 1 = 0.36 38 366726 Sy - 4 5 2(0.36 38 361 726) + 1 = 0.328602471 2 $5 " 42(0.328603471 2)+1 = 0.32185/1 807 The sequence, ( Sn ), seems to be decreasing and bounded below by 0.3 To prove ( sa) is bounded below by 0.3, apply induction : Base step : For 1 = 1 , S , # 2 > 0 . 3 Induction Hypothesis : Suppose 5 20.3 for SKEIN. Induction step : 5 k, 2 4 525 x + 1 > # 52 (0. 3) + 1 =0.316 2 2776 6 70.3 Thus, S, 30.3 for all newN, which Means ( SA) 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 70 16 9 9 4 4 induction : => 5, >52. Induction Hypothesis : Suppose SX > SKY, for IKEIN. Induction Step : SK+1 = 4 25 * +1 7 4 12ski+ 1 = 5k+2 Thus , Sn> Sny for all nell . 1 10 Q2.2 cont'ed 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 sat = lim + 4 12 sn+ 1 S = lim vas + 1 by Theorem 4:2.1( 6 ) 5 = 4 -lim (25n+1) S = - Vlim ( 25,) +1 by Theorem 4.2.1(6) 5 = 4 2lims, + 1 6y Theorem 4.2. 1 (6 ) S = 4 125 + 1 45 = 125+ 1 165 2= 25+ 1 165 2- 25- 1=0 S = --2+ ((-2)*- 4(16)(-1) by The quadratic 2 ( 16 ) formula = 2 168 32 S = 0. 3 2 0 1 7 4 1016 - 0. 17 5 1 94 1016 Since 5 >0 = 5= 0. 32019 41016 Hence, The limit of ( 5.) is 0. 3201941016