If [itex] a_n [/itex] and [itex] b_n [/itex] are convergent sequences and [itex] a_n ≤ b_n [/itex] for all n, show that a ≤ b where a and b are the limits of [itex] a_n [/itex] and [itex] b_n [/itex] respectively. since a_n and b_n are convergent, there exists an N1 such that [itex] |a_n - a| < ε [/itex] for all n > N1 and an N2 such that [itex] |bn - b| < ε [/itex] for all n > N2. I then choose N = max(N1, N2) so for all n > N, the 2 inequalities are satisfied. Since i want to show that a ≤ b, i take [itex] a < a_n + ε ≤ b_n + ε [/itex], but i am stuck here since b_n + ε is not less than b. Since this leads to a dead end, can someone give me a hint on how to approach this problem?