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Is my proof correct? Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$

Proof:

Let $a_n$ and $b_n$ be sequences such that $a_n \leq b_n \forall_n$. Suppose $\limsup a_n \nleq \limsup b_n$.

That is: $\limsup a_n > \limsup b_n$. From this we know:

$\forall_{\epsilon > 0} \exists_N \forall_{n>N} \implies |b_n - b| <\epsilon$. Where $b$ is the $\limsup b_n$.

$\forall_{\epsilon_1 > 0} \exists_{N_1} \forall_{n > N_1} \implies |a_n - a| < \epsilon_1$. Where $a$ is the $\limsup a_n$

So, let $a^* = a + \dfrac{\epsilon_1}{2}$ and let $b^* = b - \dfrac{\epsilon}{2}$.

Hence, $a^* \in |a_n - a| <\epsilon_1$ and $b^* \in |b_n - b| < \epsilon$. And clearly we see that $b^* < a^*$.Thus, we have found an element of $b_n$ namely $b^* < a^*$ an element of $a$. This is contradiction since we are given $a_n \leq b_n \forall_n$.

Therefore, the supposition is false, and $\limsup a_n \leq \limsup b_n$.

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This proof is for limits, not fot limsups –  userNaN Oct 14 '12 at 17:28
1  
How do you know that $a^*$ and $b^*$ are elements of the sequences? –  Pedro Tamaroff Oct 14 '12 at 17:29
    
Good point... How can I reformulate that piece then? –  CodeKingPlusPlus Oct 14 '12 at 17:50
    
Could I choose the minimum of the interval for b and the maximum of the interval for a. And then conclude that I have found b < a? –  CodeKingPlusPlus Oct 14 '12 at 17:52

1 Answer 1

up vote 3 down vote accepted

Recall that given a bounded sequence, we define $\limsup a_n$ as $$\lim a_n^+$$ where $$a_n^+=\sup\{a_n,a_{n+1},\dots,\}$$

Now, if $a_n\leq b_n$ for each $n$, what can you say about the relaton among each of the values:

$$a_n^+=\sup\{a_n,a_{n+1},\dots,\}$$ $$b_n^+=\sup\{b_n,b_{n+1},\dots,\}$$

What can you then say about their limits?

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Well I can say that each $b_n \geq a_n$. And then it follows that: $\lim a_n^+ \leq \lim b_n^+$ –  CodeKingPlusPlus Oct 14 '12 at 18:08
    
I mean each $a_n^+ \leq b_n^+$ And then it folows by taking the limit on both sides of the inequality: $\lim a_n^+ \leq \lim b_n^+$ –  CodeKingPlusPlus Oct 14 '12 at 18:25
    
@CodeKingPlusPlus Informally, yes. –  Pedro Tamaroff Oct 14 '12 at 18:38

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