Could someone please check my work

Prove That The sequence is Monotone and bounded . Then find The limit. Sn= 2 and s.. = 4 (5, +5) for all neIN 5 , = 2 5 2 : 7 ( 2 + 5 ) . 7 = 1.75 53 = 4 ( 7+ 5 ) = 1.6875 Sy = 4/ 1. 6875 + 5) = 1.671875 55 = 4 (1. 67 1875 + 5) = 1.667 968 75 The Sequence ( SA) seems To be decreasing and bounded below by 1.6 To prove (s,) is bounded below by 1.6 , apply induction . Bose srep : For nal, s , - 2 > 1.6 Induction Hypothesis : Suppose Sx 71.6 for 3KEIN. Induction Srep : Sky = 4 ( 5 x + 5) > # ( 1. 6 + 5 ) = 1. 65 21.6 Thus, S, > 1.6 for allnEIN, which means ( s.) is bounded below by 1.6. To prove ( sn) is decreasing and Thus monoTone, apply induction' Base Step : For n= 1, s, = 2 and s2 = 1.75 = > 5, >5 2. Induction Hypothesis: Suppose SK > 5 K, for JKEIN Induction Srep : Ski = 4 ( SK+ 5) > + (5K+1 +5) = 5k+2 Thus , Sn sati for all nEad. [1 . 6 , 2] . Hence ( Sn) is decreasing Monotone sequence bounded by By The Monotone Convergence Theorem( Theorem 4.3. 3), Since ( sn) is a bounded MonoTone Sequence, Then ( S. ) is convergent. To find the limit of ( sn) : S = lim sn S = lim smal S = lim / 4 (sn+ 5)) S= 4lim (Sn +5) by Theorem 4.2. 1( 6) S = flimsn+ 5/ by Theorem 4.2. 1( 6) 5 = + (5+ 5) 45 = 5+ 5 35 - 5 5 Hence, The- limit of ( sn) is 5