All Questions
Tagged with real-numbers proof-writing
127
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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 ...
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4
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108
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why is the co-prime part not mentioned in the definition of the rational number?
Proving $\sqrt{2}$ an irrational number is a quite popular exercise, in precalculus courses, but if we look clearly the definition that is introduced, in the beginning of the course, it never ...
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Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same. [closed]
Prove that there exist equal number of irrational numbers between any 2 rational numbers, when the difference between the 2 rational numbers is same.
If the assertion is not true then please prove ...
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7
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2
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Baby Rudin Theorem 1.19 Step 5
I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's ...
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137
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proof of infimum and upper bound of $(1+1/n)^n$
I have to prove that 2 is the minimum, and therefore infimum of the set of all numbers $(1+1/n)^n$, where $n$ are positive integer numbers. And also that it is upper bounded, not necessarily to show ...
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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 ...
- 4,227
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Prove that the closed interval [a, b] is compact.
I am studying the topology of $\mathbb{R}$ and I want to prove that the closed interval $[a,b]$ is compact using the Heine-Borel Theorem that a set in $\mathbb{R}$ is compact iff it is closed and ...
user1052225
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3
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Is a bounded polynomial constant? [duplicate]
I am trying this problem:
If $p(x)$ is a bounded polynomial for all $x\in \mathbb R$, then $p(x)$ must be a constant.
I am trying to prove it by contradition. So I assume that $p(x)$ is bounded for ...
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1
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Help With Basic Proof from Rudin PMA Chapter One
From Walter Rudin's Principles of Mathematical Analysis, Third Edition, page 20, Step 8:
I want to complete the proof of (b) by proving the following claim.
Claim: For $r\in{}\mathbb{Q}^+$ and $s\in{}...
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6
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1
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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 ...
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1
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264
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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 ...
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2
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105
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Is there a proof for triangle inequality in $\mathbb{R}$ by contradiction/absurd?
I want to prove that given $a,b,c\in\mathbb{R}$ we have $|a+b|\leq|a|+|b|$ using an absurd and reaching a contradiction.
So, I state, by absurd, that $|a+b|>|a|+|b|$, but I can't reach the ...
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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 ...
2
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If $0<a<b$ and $0<c<d$ then $bd+ac>bc+ad.$ Can this inequality be generalised to include more than just two lots of two numbers?
$0.7<2.4,\ $ and $\ 0.8< 0.9.$ By calculation, we see that $0.7\times 0.8 + 2.4\times 0.9 > 0.7\times 0.9 + 2.4\times 0.8.$
Indeed, for any pair of two numbers $\ 0<a<b\ $ and $\ 0<c&...
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Proving a rational number is NOT in the lower Dedekind cut for a transcendental number
I'm attempting to argue with a finitist who claims that transcendental numbers can't be defined as Dedekind cuts without using an infinite predicate or in some other way requiring an infinite number ...
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