This can be proved using the following facts.
(1) The limit of a subsequence of a converging sequence equals the limit of the sequence.
(2) If a_{n} and b_{n} are converging sequences and a_{n} ≤ b_{n} for all n, then lim a_{n} ≤ lim b_{n}.
This can be proved using the following facts.
(1) The limit of a subsequence of a converging sequence equals the limit of the sequence.
(2) If a_{n} and b_{n} are converging sequences and a_{n} ≤ b_{n} for all n, then lim a_{n} ≤ lim b_{n}.
The said exercise is presented in the book before defining subsequences and fact (1), so I'm not supposed to use it. That's why I was looking for a more basic proof, almost only by the definition of limit of sequence. However, I can use the following rules, in case they're helpful:
"If a sequence converges, then its limit is unique."
"Every convergent sequence is bounded."
Let A = lim aₙ and B = lim bₙ. For a given ε > 0, choose N such that for all n > N we have |aₙ - A| < ε and |bₙ - B| < ε. Pick any even n > N; then inequalities in the previous sentence imply that A < aₙ + ε and bₙ < B + ε. Therefore, A < aₙ + ε ≤ bₙ + ε < B + 2ε. Thus, for the given ε we showed that A < B + 2ε. Since this holds for any positive ε, it must be that A ≤ B. Similarly, you can show that A ≥ B.