# Thread: Limit of two sequences

1. ## Limit of two sequences

Let an and bn be two convergent sequences. Prove: if for every even n: an<=bn, and for every odd n: an>=bn, then lim an = lim bn.

Any ideas? I'm looking for a formal proof.
Thanks!

2. ## Re: Limit of two sequences

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 an and bn are converging sequences and an ≤ bn for all n, then lim an ≤ lim bn.

3. ## Re: Limit of two sequences

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."

4. ## Re: Limit of two sequences

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.

5. ## Re: Limit of two sequences

Originally Posted by loui1410
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."
Suppose that ${\lim _{n \to \infty }}{a_n} = A\;\& \;{\lim _{n \to \infty }}{b_n} = B$. Going for a contradiction, suppose that $A. Now define $\varepsilon = \frac{{B - A}}{2} > 0$.
So we have $\left( {A - \varepsilon ,A + \varepsilon } \right) \cap \left( {B - \varepsilon ,B + \varepsilon } \right) = \emptyset$.

Because of convergence, almost all the $a_n's$ are in $\left( {A - \varepsilon ,A + \varepsilon } \right)$ and almost all the $b_n's$ are in $\left( {B - \varepsilon ,B + \varepsilon } \right)$.
But that is impossible because of the alteration of odd and even terms.

6. ## Re: Limit of two sequences

emakarov, thank you - that's a great solution, I wouldn't have thought about it.
Plato's solution, however, seems more natural to me as it's similar to some other examples I've already seen.
Thank you both.