THEOREM 1:A SEQUENCE (xn) of real numbers converges to a limit a ∈ R, written\(a = \mathop {\LIM }\limits_{N \to \infty } x_n \; or \; x_n \rightarrow a \; as \; x\rightarrow \infty\)If for every ϵ > 0 there exists N ∈ N such that|xn − a| < ϵ for all n > N ⇒ a - ϵ < xn < a + ϵ for all n > NTheorem 2: If a sequence converges, then its limit is unique.Theorem 3: Let (xn) and (yn) and (zn) be sequences in R and xn ≤ yn ≤ zn for all n.If lim xn = lim zn = a, then (yn) converges to a.Proof:Take ϵ > 0. Then here are N1 and N2 so that|xn − a| < ϵ for n ≥ N1 and |zn − a| < ϵ for n ≥ N2.⇒ a − ϵ < xn < a + ϵ for n ≥ N1 and a − ϵ < zn < a + ϵ for n ≥ N2Now xn ≤ yn ≤ zn ⇒ a − ϵ < xn ≤ yn ≤ zn < a + ϵ for all n ≥ N = max {N1, N2}.This means that |yn − a| < ϵ for all n ≥ N.∴ Sequence (yn) converges and lim (xn) = lim (yn) = lim (zn)