All Questions
10
questions with no upvoted or accepted answers
3
votes
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answers
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My first simple direct proof (very simple theorem on real numbers). Please mark/grade.
What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Let $I = [a,b]$ be a non-empty closed ...
- 1,321
1
vote
0
answers
60
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Exercise 6, Section 2.2 of Hoffman’s Linear Algebra
(a) Prove that the only subspaces of $\Bbb{R}$ are $\Bbb{R}$ and the zero subspace.
(b) Prove that a subspace of $\Bbb{R}^2$ is $\Bbb{R}^2$, or the zero subspace, or consist of all scalar multiples of ...
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1
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0
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59
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Real Analysis - Prove that there exists $n, m \in \Bbb{N}$ such that $2m\pi + \frac{\pi}{2} - \epsilon < n < 2m\pi + \frac{\pi}{2}$.
This is a claim that I had made while finding the supremum of $(\sin(n))_{n \in \Bbb{N}}$. The supremum is $1$ if there exists $n \in \Bbb{N}$ such that for all $\epsilon' > 0$, $1-\epsilon' < \...
- 4,335
1
vote
0
answers
67
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Proof: For any subsequence $a_{n_k}$ Prove $\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n$
For any convergent subsequence $a_{n_k}$ of $a_n$, Prove: $$\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n.$$
My attempt
For this proof it should be noted that $...
- 70
1
vote
0
answers
1k
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Let $a < b $ be real numbers and consider set T = $\mathbb Q \ \cap \ [a,b].$ Show $\sup \ T =b$
Let $a < b $ be real numbers and consider set T = $\mathbb Q\cap [a,b].$ Show $\sup T =b$
I needed help checking if my proof is correct. If it isn't correct can you please provide the correct ...
- 3,306
1
vote
0
answers
80
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How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
$1.$ Every subsequence of $(s_n)$ has a further subsequence that ...
- 2,756
0
votes
0
answers
137
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Prove $(-a)^{-1} = -a^{-1}$
This is my proof for part (17) of Lemma 2.3.2 in Bloch's Real Analysis. I'd like if someone verifies that I did not miss or skip a step, did anything unjustified, or anything of this sort. I will ...
0
votes
1
answer
35
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Example of basis
Excuse me , can you see this question
, the collection of all open intervals (a,b) together with the one-point sets {n} for all positive and negative integers n is a base for a topology on a real ...
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0
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20
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Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
I am trying to formally prove:
$ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
where n is an integer, and x and y are natural numbers.
It is obvious that, when $\frac xy$ is ...
- 101
0
votes
1
answer
104
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$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)
As an exercise, I tried to prove the following theorem.
Please share your thoughts about what I wrote.
(The proof only uses the utensils which are listed below.)
Theorem
Let $x,y,z \in \mathbb{R}$.
...
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