# Proof of Sohrab's Proposition 2.2.39 (a) | Upper & Lower Limits

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In summary: BUT ...... how do we justify treating a real number (viz. a) like a sequence ...... that is how is it valid that a = (a,a,a, ... ... )
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with the proof of Proposition 2.2.39 (a)Proposition 2.2.39 (plus definitions of upper limit and lower limit ... ) reads as follows:
View attachment 9243

Can someone please demonstrate a formal and rigorous proof of Part (a) of Proposition 2.2.39 ...Help will be appreciated ... ...

Peter

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I think I covered (a) at the end of my post
https://mathhelpboards.com/analysis-50/upper-lower-linits-lim-sup-lim-inf-sohrab-proposition-2-2-39-b-26469-new.html

where I said "This is another basic property of limits"

you just need to re-label one of the sequences to be a or b... (which is an isolated point / peculiar sequence but it doesn't change any of the intuition or epsilon management)

steep said:
I think I covered (a) at the end of my post
https://mathhelpboards.com/analysis-50/upper-lower-linits-lim-sup-lim-inf-sohrab-proposition-2-2-39-b-26469-new.html

where I said "This is another basic property of limits"

you just need to re-label one of the sequences to be a or b... (which is an isolated point / peculiar sequence but it doesn't change any of the intuition or epsilon management)
Hi steep ...

Thanks for the help ,,, but ... having trouble following you on this one ...

Sorry to be slow ... but can you give more details of the proof for (a) ...

Thanks again fr your help ...

Peter

so using $u_n$ as the inf sequence, we know

$a \leq u_n$ for all $n$ and we know (monotone convergence) that $u_n \to L_u \geq a$ I'm going to reuse that other argument almost verbatim to make a point
====
(This is another basic property of limits... if you aren't familiar, argue by contradiction that $a - L_u = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$

you have $\vert u_n - L_u\vert \lt \epsilon$ and by optional analogy with before $\vert a - a\vert =0\lt \epsilon$ but this implies $a \gt u_n$ which is a contradiction -- sketching this out is best... it implies that $u_n \lt L_u + \frac{1}{10}c \lt L_u + \frac{9}{10}c = L_u + c - \frac{1}{10}c = a- \frac{1}{10}c \lt a$

but we are told $a \leq u_n$ holds for all n

steep said:
so using $u_n$ as the inf sequence, we know

$a \leq u_n$ for all $n$ and we know (monotone convergence) that $u_n \to L_u \geq a$ I'm going to reuse that other argument almost verbatim to make a point
====
(This is another basic property of limits... if you aren't familiar, argue by contradiction that $a - L_u = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$

you have $\vert u_n - L_u\vert \lt \epsilon$ and by optional analogy with before $\vert a - a\vert =0\lt \epsilon$ but this implies $a \gt u_n$ which is a contradiction -- sketching this out is best... it implies that $u_n \lt L_u + \frac{1}{10}c \lt L_u + \frac{9}{10}c = L_u + c - \frac{1}{10}c = a- \frac{1}{10}c \lt a$

but we are told $a \leq u_n$ holds for all n

Thanks again for your help steep ...

After reflecting on what you have written ...

... I think the essence of your suggested proof of $$\displaystyle a \leq \underline{lim} (x_n)$$ is as follows ...... now ... $$\displaystyle u_n = \text{inf} \{ x_k \ : \ k \geq n \}$$ and $$\displaystyle a \leq u_n \leq b$$ ...

If we accept that $$\displaystyle (u_n)$$ is increasing (or at least non-decreasing) then given that $$\displaystyle (u_n)$$ is also bounded above by b, we then have, by the Monotone Convergence Theorem, that $$\displaystyle (u_n)$$ is convergent ...... that is $$\displaystyle (u_n) \longrightarrow L_u$$ where $$\displaystyle L_u$$ is some real number ... and of course $$\displaystyle L_u = \underline{lim} (x_n)$$Then ... to prove $$\displaystyle a \leq \underline{lim} (x_n)$$ ... I think you do something like the following ...

... let $$\displaystyle a = (a,a,a, ... ... )$$ and so given that $$\displaystyle a \leq u_n \ \forall \ n \in \mathbb{N}$$ we have ...

... $$\displaystyle \lim_{ n \to \infty } (a) \leq \lim_{ n \to \infty } (u_n )$$

$$\displaystyle \Longrightarrow a \leq \underline{lim} (x_n)$$ ... BUT ...... how do we justify treating a real number (viz. $$\displaystyle a$$) like a sequence ...

... that is how is it valid that $$\displaystyle a = (a,a,a, ... ... )$$
Is the above correct ...

Can you help further ... ?

Peter

Peter said:
...
Then ... to prove $$\displaystyle a \leq \underline{lim} (x_n)$$ ... I think you do something like the following ...

... let $$\displaystyle a = (a,a,a, ... ... )$$ and so given that $$\displaystyle a \leq u_n \ \forall \ n \in \mathbb{N}$$ we have ...

... $$\displaystyle \lim_{ n \to \infty } (a) \leq \lim_{ n \to \infty } (u_n )$$

$$\displaystyle \Longrightarrow a \leq \underline{lim} (x_n)$$ ... BUT ...... how do we justify treating a real number (viz. $$\displaystyle a$$) like a sequence ...

... that is how is it valid that $$\displaystyle a = (a,a,a, ... ... )$$
Is the above correct ...

Can you help further ... ?

Peter

That is the essence of it. The underlined part is what worries me because it is technically a fine question, but I inadvertently created this as an issue... because I wanted to stress that the intuition (and epsilon management) is basically the same as in your other question. Now kindly forget the fact that I made that analogy. And let's take another look at the problem.

Consider $u_n$ as a sequence. and $a$ is just some constant real number that is a lower bound on $u_n$. How do I know that $a \leq u_n$ for all n? because your book tells me this is true for the specific sequence involved.

so copying and pasting from what I wrote before, with minor adjustments (to make the point that the epsilon management is about the same) I would say:

= = = = =
argue by contradiction that $a - L_u = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$

you have $\vert u_n - L_u\vert \lt \epsilon$ but this implies that

$u_n \lt L_u + \frac{1}{10}c \lt L_u + c = a$

but we are told $a \leq u_n$ holds for all n which is a contradiction

Last edited:
steep said:
That is the essence of it. The underlined part is what worries me because it is technically a fine question, but I inadvertently created this as an issue... because I wanted to stress that the intuition (and epsilon management) is basically the same as in your other question. Now kindly forget the fact that I made that analogy. And let's take another look at the problem.

Consider $u_n$ as a sequence. and $a$ is just some constant real number that is a lower bound on $u_n$. How do I know that $a \leq u_n$ for all n? because your book tells me this is true for the specific sequence involved.

so copying and pasting from what I wrote before, with minor adjustments (to make the point that the epsilon management is about the same) I would say:

= = = = =
argue by contradiction that $a - L_u = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$

you have $\vert u_n - L_u\vert \lt \epsilon$ but this implies that

$u_n \lt L_u + \frac{1}{10}c \lt L_u + c = a$

but we are told $a \leq u_n$ holds for all n which is a contradiction
Thanks for all your help on this problem, steep ...

... it is is much appreciated ...

Peter

## 1. What is Sohrab's Proposition 2.2.39 (a)?

Sohrab's Proposition 2.2.39 (a) is a mathematical theorem that states the upper and lower limits of a sequence are equal if and only if the sequence is convergent.

## 2. How is this proposition useful in mathematics?

This proposition is useful in various mathematical fields, such as calculus and analysis, as it provides a way to determine the convergence of a sequence. It also helps in proving other theorems and solving mathematical problems.

## 3. Can you explain the upper and lower limits in more detail?

The upper limit of a sequence is the largest number that the terms of the sequence approach as the index approaches infinity. The lower limit is the smallest number that the terms of the sequence approach as the index approaches infinity.

## 4. Can you provide an example of Sohrab's Proposition 2.2.39 (a)?

For example, let's consider the sequence {1/n}. The upper limit of this sequence is 0, as the terms get closer and closer to 0 as n approaches infinity. The lower limit is also 0, as the terms will never be smaller than 0. Therefore, according to Sohrab's Proposition 2.2.39 (a), this sequence is convergent.

## 5. Are there any other variations of Sohrab's Proposition 2.2.39?

Yes, there are other variations of this proposition, such as Proposition 2.2.39 (b) which states that the limit of a sequence is equal to the upper and lower limits if and only if the sequence is convergent. There are also similar propositions for sequences with infinite limits and for sequences in metric spaces.

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