Could someone please check my work

Prove that each sequence is monotone and bounded. Then find the limit. 1. 81 " 1 and $4 . 1 - 7 (3. + 5) for n E N. . Prove that s. is monotone: S = 1 and an + 1 - 7 (8x + 5) for all n E N $1 = 1 $2 = (1 + 5) = 1.5 83 = (1.5 + 5) = 1.625 81 = - (1.625 + 5) = 1.65625 85 - (1.65625 + 5) = 1.6640625 86 = (1.6640625 + 5) = 1.666015625 The sequence, (Sn), seems to be increasing and bounded by [1, 1.67]. To prove (Sn) is increasing and thus monotone by definition 4.3.1, apply induction: Base step: For n = 1 , $1 = 1 and $2 = 1.5 = $1