Upper and Lower Linits (lim sup and lim inf) - Sohrab Proposition 2.2.39 (b) ....

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In summary: 's monotone convergence theorem states that if $u_n$ is a monotone non-decreasing sequence and the sequence is bounded, then $u_n \lt v_n$ for all $n$.
  • #1
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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 (b)Proposition 2.2.39 (plus definitions of upper limit and lower limit ... ) reads as follows:
View attachment 9244
Can someone please demonstrate a formal and rigorous proof of Part (b) of Proposition 2.2.39 ...Help will be appreciated ... ...

Peter
 

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  • #2
I presume you know monotone convergence theorem...

$u_n$ is a monotone non-decreasing sequence and the sequence is bounded, so it has a limit $L_u$
$v_n$ is a monotone non-increasing sequence and is bounded so it has a limit $L_v$

but for all $n$
$u_n \leq v_n$
by construction hence passing limits,
$L_u \leq L_v$
(This is another basic property of limits... if you aren't familiar, argue by contradiction that $L_u - L_v = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$ in each sequence (a slightly more careful approach is $N =\max\big(N_v, N_u\big)$)

you have
====
$\vert u_n - L_v\vert \lt \epsilon$
edit: to cleanup a typo, this should have said: $\vert u_n - L_u\vert \lt \epsilon$ . I had the wrong subscript.
====
and $\vert v_n - L_v\vert \lt \epsilon$ but this implies $u_n \gt v_n$ which is a contradiction -- sketching this out is best... it implies that $v_n \lt L_v + \frac{1}{10}c \lt L_v + \frac{9}{10}c = L_v + c - \frac{1}{10}c = L_u- \frac{1}{10}c \lt u_n$ )
 
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  • #3
steep said:
I presume you know monotone convergence theorem...

$u_n$ is a monotone non-decreasing sequence and the sequence is bounded, so it has a limit $L_u$
$v_n$ is a monotone non-increasing sequence and is bounded so it has a limit $L_v$

but for all $n$
$u_n \leq v_n$
by construction hence passing limits,
$L_u \leq L_v$
(This is another basic property of limits... if you aren't familiar, argue by contradiction that $L_u - L_v = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$ in each sequence (a slightly more careful approach is $N =\max\big(N_v, N_u\big)$)

you have $\vert u_n - L_v\vert \lt \epsilon$ and $\vert v_n - L_v\vert \lt \epsilon$ but this implies $u_n \gt v_n$ which is a contradiction -- sketching this out is best... it implies that $v_n \lt L_v + \frac{1}{10}c \lt L_v + \frac{9}{10}c = L_v + c - \frac{1}{10}c = L_u- \frac{1}{10}c \lt u_n$ )
Thanks for the help, steep ...

Just reflecting on what you have written...

Peter
 

1. What is the definition of lim sup and lim inf?

Lim sup and lim inf are two important concepts in mathematical analysis that are used to describe the behavior of a sequence or a function. Lim sup (limit superior) is the largest limit point that a sequence or function can approach, while lim inf (limit inferior) is the smallest limit point that a sequence or function can approach.

2. How are lim sup and lim inf related to upper and lower limits?

In Sohrab Proposition 2.2.39 (b), it is proven that for a sequence or a function, the lim sup is equal to the upper limit, and the lim inf is equal to the lower limit. This means that lim sup and lim inf are closely related to upper and lower limits, and they provide a more precise description of the behavior of a sequence or a function.

3. What is the significance of lim sup and lim inf in mathematical analysis?

Lim sup and lim inf are important tools in mathematical analysis as they help to determine the convergence or divergence of a sequence or a function. They also provide information about the behavior of a sequence or a function near its limit points, which can be useful in various applications in mathematics and other fields.

4. How are lim sup and lim inf calculated?

The calculation of lim sup and lim inf depends on the specific sequence or function being analyzed. In general, lim sup is calculated by finding the largest limit point that the sequence or function approaches, while lim inf is calculated by finding the smallest limit point. This can be done using various techniques such as the squeeze theorem, the monotone convergence theorem, and the Bolzano-Weierstrass theorem.

5. Can lim sup and lim inf be equal to each other?

No, lim sup and lim inf can never be equal to each other. This is because lim sup represents the largest limit point that a sequence or function can approach, while lim inf represents the smallest limit point. If these two values were equal, it would mean that the sequence or function has only one limit point, which is not possible in most cases.

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