Could someone please check my work

Let ( a. ) be the sequence defined by a. = (1 + - ) . The limit of (s. ) is referred to as e and is used as the base for natural logarithms. The approximate value of e is 2.71828. 1. Use the binomial theorem (Exercise 3.1.28) to show that (s.) is an increasing sequence with an an which is equivalent to showing an+ ! 1 (1+ 4 ) " END THAT ATAT (10+ 1-4) (MAT)" ) (1 + # ) END TO-B TAT (1"-4 ) (#) " ) S1+1 + (24 7) since Kel S 24-1 (proved by induction below) (m + 1) . 2m-1 ER ! ( RT ) (1) ( 1 - #) (1 -727).. ( 1 - 4-4) Since (m + 1) > 2 , ( m + 1 ) . 2m - 1 2 2 . 2m -1 Ex-O (x) (1) ( 1 - #) (1 - 2). ( 1 - 4 1) cancelling terms = 2m Thus, (m + 1)! > 2m 2 1 since by the Archimedean property (Theorem 3.3.10(c)), The induction step holds. 1 1 since n + 1 2 n then n + In Therefore, (k!) > 2k- 1 Thus, each term in the numerator after the term, ( 37 ) is greater than the respective term in the denominator, making the numerator larger than the denominator. Since the numerator is larger than the denominator, the quotient is greater than 1. Thus, Sn + 1 > Sn for all n 2 2 E N, which means (Sn) is an increasing sequence by 2. Conclude that (Sn) is convergent. definition 4.3.1. When n = 1 , 8n = ( 1 + 2 ) = $1 = (1+ 4) - 2. so since (8,) is increasing, MEN. Sn 2 2 , which by the definition of boundedness means, S, is bounded below by 2 for all WTS Sn