Converges to l. example: prove directly from the definition that the sequence n+3 a, nen an 2n+1 is a cauchy sequence. solution: let s> 0 be given. let n be a p

Converges to L. Example: Prove directly from the definition that the sequence n+3 a, neN an 2n+1 is a Cauchy sequence. Solution: Let s> 0 be given. Let N be a positive integer to be chosen. Suppose that n, m 2 N. We have m+3 m-n a,--3 2n+1 2m+1 (2n+1)(2m+1) (2n+1)+ (2m+1) -2 (2n+1)(2m+1) 2m+2n (2n+ 1)(2m+1) 1 1 2 2m+1 2n+1 (2n +1)(2m +1) 1 2m+1 2n+1 1