All Questions
38
questions
1
vote
1
answer
54
views
My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$
Is it reasonable to prove the following (trivial) theorem?
If yes, is there a better way to do it?
Let $x, y \in \mathbb{R}$.
Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$.
$\textbf{...
- 399
0
votes
1
answer
104
views
$xy \le xz$ 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 $x,y,z \in \mathbb{R}$.
...
- 1,321
3
votes
1
answer
2k
views
$x^n y^n = (xy)^n$, 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
\begin{equation*}
x^n y^n ...
- 1,321
1
vote
2
answers
4k
views
An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$
Problem:
Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon &...
- 6,684
2
votes
3
answers
75
views
If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof
For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).
Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.
Assume, x, y ∈ ℝ # Domain assumption
...
- 93
2
votes
1
answer
398
views
My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.
What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style.
Theorem
Among three elements of ...
- 1,321
3
votes
0
answers
190
views
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
1
answer
125
views
Possible book correction or am I missing something?
Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
- 825