Post History
#4: Post edited
by
Snoopy
·
2024-07-17T13:24:06Z (4 months ago)
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- > To avoid tedious discussion with infinity, assume in addition that both $a_n$ and $b_n$ are bounded sequences.
- **Notes.** The question is based on the Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.So, the question reduces to showing the reversed inequality:- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
There is one "_algebraic_" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- > To avoid tedious discussion with infinity, assume in addition that both $a_n$ and $b_n$ are bounded sequences.
- **Notes.** The question is based on the Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- (See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.)
- Thus, the question reduces to showing the reversed inequality:
- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- Based on a few basic observations of the definition of liminf, there is one "_algebraic_" way to show this inequality, which conceals the use of many quantifiers. I will write my own answer below. Alternatives are welcome.
#3: Post edited
by
Snoopy
·
2024-01-09T13:27:05Z (11 months ago)
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- > To avoid tedious discussion with infinity, assume in addition that both $a_n$ and $b_n$ are bounded sequences.
**Notes.** The question is based on this Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.
- So, the question reduces to showing the reversed inequality:
- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
There is one "algebraic" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- > To avoid tedious discussion with infinity, assume in addition that both $a_n$ and $b_n$ are bounded sequences.
- **Notes.** The question is based on the Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.
- So, the question reduces to showing the reversed inequality:
- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- There is one "_algebraic_" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.
#2: Post edited
by
Snoopy
·
2024-01-09T01:36:53Z (11 months ago)
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- **Notes.** The question is based on this Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.
- So, the question reduces to showing the reversed inequality:
- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- There is one "algebraic" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.
- > **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
- $$
- \liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
- $$
- > To avoid tedious discussion with infinity, assume in addition that both $a_n$ and $b_n$ are bounded sequences.
- **Notes.** The question is based on this Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
- It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
- $$
- \liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.
- So, the question reduces to showing the reversed inequality:
- $$
- \liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
- $$
- There is one "algebraic" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.
#1: Initial revision
by
Snoopy
·
2024-01-09T01:33:42Z (11 months ago)
$\liminf (a_n+b_n) = \liminf(a_n)+\liminf(b_n)$ provided that $\lim a_n$ exists
> **Question.** Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
$$
\liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n).
$$
**Notes.** The question is based on this Wikipedia article on [limit inferior and limit superior](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior), where the statement above is mentioned explicitly in the article without any proof.
It follows immediately from the [definition](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Definition_for_sequences) of $\liminf$ that
$$
\liminf_{n\to\infty}(a_n+b_n)\geq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
$$
See [this section](https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior#Properties) of the article.
So, the question reduces to showing the reversed inequality:
$$
\liminf_{n\to\infty}(a_n+b_n)\leq\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)
$$
There is one "algebraic" way to show this inequality, which I will write as my own answer below. Alternatives are welcome.