All Questions
38
questions
0
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1
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22
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How to more formally prove this inequality
This is a simple problem I came up with while doing another problem:
Given: $n < (n + \frac{1}{2}) < y < (n + 1)$
Prove: $|y - n| > |y - (n + 1)|$
So how I proved it was simply using the ...
- 865
1
vote
0
answers
59
views
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 ...
- 4,227
6
votes
1
answer
202
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Different definitions of the archimedean property
In some textbooks I have seen the archimedean property defined as:
for some positive real $x$, real number $y$, there exists a natural $n$ such that $nx>y$.
In other textbooks the archimedean ...
0
votes
1
answer
265
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Proving trichotomy and transitivity from the definition of an ordered field
I'm reading a set of notes (and can provide the link if anyone is interested) which attempt to build up the properties of an ordered field. After defining a field based on its axioms, it defines an ...
- 1,418
0
votes
0
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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 ...
1
vote
0
answers
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
0
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1
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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 ...
- 93
2
votes
1
answer
86
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Alternative proof of $a\times0= 0$
I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
2
votes
1
answer
319
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Bad Proof? Between any two reals is a rational number
I know about the proof found here: Proof there is a rational between any two reals.
I wanted to know if this similar proof is also correct?
Assume $x > 0$. Since $y > x$, it follows $y-x>0$....
1
vote
0
answers
67
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Proof: For any subsequence $a_{n_k}$ Prove $\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n$
For any convergent subsequence $a_{n_k}$ of $a_n$, Prove: $$\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n.$$
My attempt
For this proof it should be noted that $...
- 70
3
votes
1
answer
128
views
Exercise about sub-$\sigma$-algebra of $\mathcal{B}(\mathbb{R})$
Let $C=\{(-a, a): a \in \mathbb{R}\}$ and $F=\sigma(C)$.
Prove that $F=\mathcal{B}(\mathbb{R})\cap\{A\subseteq\mathbb{R}: A=-A\}$.
I don't have problems in proving $F\subseteq \mathcal{B}(\mathbb{R})\...
- 319
1
vote
2
answers
348
views
Explain this "contradiction" of the proof of $x > 0$ iff $x \in \mathbb{R}^{+}$
I am working through Apostol's Calculus I and just read about the order axioms. He presents a new undefined concept called positiveness, gives the axioms and then defines symbols $<, >, \leq, \...
4
votes
2
answers
12k
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Proof clarification - If $ab = 0$ then $a = 0$ or $b =0$
I came across a proof for the following theorem in Apostol Calculus 1. My question is regarding (1) in the proof, why is this part necessary? I don't see why you can't begin with (2)
Theorem 1.11
If ...
0
votes
2
answers
92
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Validity of Proof for 'Possibility of Subtraction' from Apostol 1
I attempted a proof before reading the solution Apostol provides. I don't think it is valid but I am trying to determine why
Theorem 2, Possibility of Subtraction: Given $a \text{ and } b$, there is ...
3
votes
2
answers
69
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Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.
I came across the implication
$$x < y \Rightarrow x^n < y^n$$
$$x,y>0, n\in Z^+$$
in a textbook and came up with the following proof.
Proof
Since $x<y$ the following chain of inequalities ...
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