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## Homework Statement

Consider three sequences a(n), b(n) and s(n) such that an a(n) <= s(n) <= b(n) for all n and lim(a(n)) = lim(b(n)) = s. Prove that lim(s(n)) = s

(n) is a subscript

## Homework Equations

the book i am doing this problem from is elementary analysis, the theory of calculus and it defines the definition of a limit for a sequence as

|s(n) - s| < E

so its sort of like an epsilon delta proof

## The Attempt at a Solution

i really dont know how to do this problem. i tried to do it, but was totally wrong so i was wondering if you guys could help me understand the solution. i wont post the whole thing, just the solution up to the part that i didnt understand.

book proof

let E > 0. to prove this, we need to show s-E < s(n) < s + E for large n. since lim (a(n)) = s, there exists N(1) such that |a(n) - s| < E for n > N(1). in particular

n > N(1) implies s - E < a(n)

this is the part that i dont get. i cant figure out |a(n) - s| < E and n > N(1) implies that. the only way that makes sense is if we said that (a(n) - s) < 0 therefore |a(n) - s| = - (a(n)-s) and then we can get what n > N(1) implies. however, if this is true, why can we assume a(n) - s < 0 or am i just misunderstanding what the thing is saying