Is this correct

[primal mesa]

I think I have found an answer please tell waht mistakes I am doing

[primal mesa]

Let us take Set A and Set B such that Set A = [a,b] and Set B = [c,d]
So Set A $\cup$ Set B = [a,d] and this is equal to (a+1 , d+1)

[pulsar trail]

Someone_Random

[scarlet pier]

If $b < c$. then $A\cup B = [a,b] \cup [c,d] \subsetneq [a,d]$

[pulsar trail]

vin100

[scarlet pier]

,,[a,d] \ne (a + 1, d+ 1)

[pulsar trail]

vin100

[scarlet pier]

just consider the real number a to see why

[primal mesa]

ahh ok

so could you give me a hint?

[scarlet pier]

You have one 🍕. When you divide it equally into n portions, each portion is 1 / n of the original 🍕. The greater the n, ...

,tikz[scale=10]
\pgfmathsetmacro\myN{100}; % last term to be plotted
\draw[->] (-0.1,0) -- (1.1,0);
\node[label=below:0] at (0,0) {$\times$};
\node[label=below:1] at (1,0) {$\times$};
% labelled sequence points of 1/n
\foreach \x in {1/1, 1/2, 1/..., 1/4} \node[label={[font=\scriptsize, red!60!yellow]above:$\x$}, minimum size=0.5pt, inner sep=2pt, fill=red!70!yellow, circle] at (\x,0) {};
% unlabelled sequence points of 1/n
\foreach \x in {1/5, 1/6, 1/..., 1/\myN} \node[minimum size=0.5pt, inner sep=0.7pt, fill=red!60!yellow, circle] at (\x,0) {};
% diagram title
\node[above] at (current bounding box.north) {first $\color{red!60!yellow}N = \myN$ terms in sequence $\left(\frac1n\right)_{n \in \bN}$};

[pulsar trail]

vin100

[sleek ember]

nah bruh how do you just do that with a bot

idk where to even get started to learn to help using bot

[scarlet pier]

,help

[pulsar trail]

A brief description and guide on how to use me was sent to your DMs!
Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!

[scarlet pier]

#resources message

code can be copied and tweaked from any TikZ tutorials for beginners

[primal mesa]

the smaller 1/n is

[scarlet pier]

think: in the above diagram, if I increase N to 1000...000, what would be the distance between 0 and the leftmost red dot?

[pulsar trail]

Deadeye

[sleek ember]

=0

[primal mesa]

0

[scarlet pier]

pratically yes, theoretically no

[scarlet pier]

now you may imagine that this is an animation with N increasing from 1 to 1000.....000. how much 1 / n portion of🍕 would you get at the end?
answer: ||it's always >0, but "pratically it's nothing". try to capture this verbal intuition with symbols||

[primal mesa]

it will be 0.0000000..1

[scarlet pier]

let's write it in a more rigorous way: it's smaller than

• 0.1
• 0.001
• 0.000001
• 0.00000001
• ...

[scarlet pier]

imagine each rot dot become a closing parenthesis )
take any negative number a and draw [a, 1 / n) on the number line.
what do you find?

[primal mesa]

do we have to use limits?

[scarlet pier]

❌

it's just a basic lemma in arithmetic

when my prof in intro Real Analysis course went over this, he said: "rmb this. it will be very useful in the future"

[primal mesa]

it should less than some epsilon such that 0 < 1/n < epsilon

[scarlet pier]

in your original question, the hint mentions "arbitrary intersection of open intervals". try following this hint, and think about what would happen to

$$\cap\limits_{n \in \bN} \Big[ a, \frac1n \Big).$$

[pulsar trail]

vin100

$$\cap\limits_{n \in \bN} \Big[ a, \frac1n \Big).$$
```Compilation error:```! Limit controls must follow a math operator.
l.106 $$\cap\limits
                   _{n \in \bN} \Big[ a, \frac1n \Big).$$
I'm ignoring this misplaced \limits or \nolimits command.

Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
(./701077747887243354.aux)
 ***********
LaTeX2e <2023-11-01>
L3 programming layer <2020/03/25>```

[scarlet pier]

,,\bigcap_{n \in \bN} \Big[ a, \frac1n \Big).

[pulsar trail]

vin100

[primal mesa]

hey i forgot what the big intersection does , could mind to explain\

[scarlet pier]

you know what A ∩ B means?

[primal mesa]

yea

[scarlet pier]

take a collection of two sets $\mathcal{C} = {A, B}$. We can write $A\cap B = \cap\mathcal{C}$

[pulsar trail]

vin100

[scarlet pier]

$\cap\mathcal{C}$'' can be elaborated as $\bigcap_{S \in \mathcal{C}} S$''.

[pulsar trail]

vin100

[primal mesa]

so s belongs to A and B right?

[scarlet pier]

1. variable names are cAsE sensitive. (take f and its antiderivative F for example)
2. S ∈ {A, B}, S ∉ A, ...

[primal mesa]

ahh ok

[scarlet pier]

now think about $\mathcal{C}$ as a collection of three sets, say $\mathcal{C} = {S_1, S_2, S_3}$.

[pulsar trail]

vin100

[scarlet pier]

,,\cap\mathcal{C} = \bigcap_{S \in \mathcal{C}} S

[pulsar trail]

vin100

[scarlet pier]

it's all the intersection of all sets belonging to $\mathcal{C}$

[pulsar trail]

vin100

[scarlet pier]

write this in $\forall$'' and $\exists$'' quantifiers

[pulsar trail]

vin100

[scarlet pier]

,,x \in \cap\mathcal{C} \iff \left(\forall S \in \mathcal{C} : x \in S \right)

[pulsar trail]

vin100

[primal mesa]

ok

[scarlet pier]

here you see that the three sets are indexed by $I = {1, 2, 3}$.

[pulsar trail]

vin100

[scarlet pier]

so the writing $\bigcap_{i \in I} S_i$'' is more often, as you don't need to define $\mathcal{C} = \{S_i \mid i \in I\}$ in order to write $\cap\mathcal{C}$''.

[pulsar trail]

vin100

[scarlet pier]

but i'm covering this "intersection sign followed by the name of a collection of sets" for cultural reasons.
some topology textbooks show an arbitrary intersection like this (to save ink)

besides, $\cap\varnothing$'' is a nice abbreviation for $\bigcap_{i \in \varnothing} S_i$''.

[pulsar trail]

vin100

[primal mesa]

ahh ok

[scarlet pier]

so the above countable intersection of intervals

[scarlet pier]

can be interpreted

[scarlet pier]

using this fact

with $I = \bN$ as the \textbf{index set}

[pulsar trail]

vin100

[primal mesa]

ok

[scarlet pier]

the basic lemma is "a nonnegative number smaller than all positive number is zero".

Lemma Let a ≥ 0. For all b > 0, 0 ≤ a ≤ b. Then a = 0.
Proof: Straightforward by contradiction.

[primal mesa]

ok so how couuld we use this in the problem? Is it if we take the smallest closed set and prove it can be open will that work?

[scarlet pier]

lemme a give brief topological explanation why this won't work

at the end

it has sth to do with connectedness

[scarlet pier]

you need to develop a symbolic instinct between "intersection" and "for all"

rmb that in naïve set theory, to show set equalities (say (A ∩ B) ∩ C = A ∩ (B ∩ C)), one might write sth like
x ∈ (A ∩ B) ∩ C ⟺ x ∈ A ∩ B AND x ∈ C ⟺ ⋯ ⟺ x ∈ A ∩ (B ∩ C)

Since the above Lemma concerns elements, you need to write the countable intersection like

,,x \in \bigcap_{n \in \bN} \Big[ a, \frac1n \Big) \iff \cdots

[pulsar trail]

vin100

[primal mesa]

$\forall$n \in \mathbb{N}$

[pulsar trail]

Someone_Random
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[scarlet pier]

,,\forall n \in \bN

[pulsar trail]

vin100

[primal mesa]

wait how are we using this is the proof?

[scarlet pier]

continue with

[scarlet pier]

this one

oh it should be the Archmedian property of integer to be used here

[primal mesa]

wait if I am correct Archimeadan property says $a,b \in \mathbb{R}$ . $\exists n$ such that a<nb

[pulsar trail]

Someone_Random

[scarlet pier]

✅

try to interpret this verbally. for any given a, n can be as large has you want so that it's greater than a.

so ...

[scarlet pier]

you need to capture this intuition]

sorry i gtg

[pulsar trail]

vin100

[scarlet pier]

ofc u can use limit but for an introductory math analysis course, that would be overkill

your question wants ... so ||make a claim that x is not positive. proof by contradiction. end game||

[primal mesa]

so n must be larger than a?

[scarlet pier]

draw a picture, and you'll see. the question askes for intersecting intervals. what if a ≥ 0 with shrinking 1 / n? what would the intersection be? would that respond to the question?

[primal mesa]

we will have p such that $p \in [0,\infty)$

[pulsar trail]

Someone_Random

[scarlet pier]

❔

[primal mesa]

the intersection of [a,1/n) will be p such that p belogn to [0,infinity)

[scarlet pier]

you can be more precise, so that your conclusion answers the question.

btw i assumed a < 0, so the countable intersection above shouldn't be a singleton

[primal mesa]

could you give a hint ?

[scarlet pier]

hint already given just capture the instinct

1 / n of a pizza. as n grows larger and larger, you pratically gets nothing

[scarlet pier]

if you find things difficult to understand, you can try drawing a picture

of some first few intervals for [a, 1 / n)

[primal mesa]

if p is the intersection of the interval then $p \in [a,1/n)$

[pulsar trail]

Someone_Random

[scarlet pier]

even though your "if ... then ..." statement is logically true, that doesn't mean anything cuz the premise is wrong. the countable intersection above is, by construction, NOT a singleton. The closed interval [a, a / 2] is contained inside this countable intersection.

[primal mesa]

what do you mean I could not understand

[scarlet pier]

what don't you understand?

just draw a picture, and you'll see

[primal mesa]

ok

[primal mesa]

if $x \ge 1$ then $\bigcap [a,1/n) = \Phi$ and if $a \le 1$ then $\bigcap [ a,1/n) \in [a,1/n)$

[pulsar trail]

Someone_Random

[scarlet pier]

you might want to use the cases environment, like

,,|x| = \begin{cases} x &\qif x \ge 0 \ -x & \qotherwise \end{cases}

[pulsar trail]

vin100

\begin{gather*}
|x| = \begin{cases} x &\qif x \ge 0 \\ -x & \qotherwise \end{cases}
\end{gather*}

[scarlet pier]

the writing of $\cap[a,1/n)$'' is problematic cuz the index set is absent. the desired countable intersection shld b $\bigcap_{n \in \bN} [a, 1/n)$'' instead.

[pulsar trail]

vin100

the writing of ``$\cap[a,1/n)$'' is problematic cuz the index set is absent.  the desired countable intersection shld b ``$\bigcap_{n \in \bN} [a, 1/n)$'' instead.

[scarlet pier]

your use of '=' is incorrect. '=' is a binary relation, so there shld b a math quantity on its right

the condition ``$a \le 1$'' is unnecessary, cuz we've set, by construction, $a<0$, so that the desired countable intersection is NOT a singleton

[pulsar trail]

vin100

[scarlet pier]

but it's an interval (with nonzero interval length) instead

the writing of $\cap[a, 1/n) \in [a, 1/n)$'' is incorrect. apart from index set in the countable intersection, the intersection is a subset of $\bR$, and you can't say a set belongs to another set''. instead, you can say

[pulsar trail]

vin100

[scarlet pier]

• "an element of a set belongs to another set", or
• "a set is a subset (or superset) of another set"

[primal mesa]

ok

[primal mesa]

So is this correct :
$I = [0 , 1 - \frac{1}{n}]$ for $ n = 1,2,3,4,5,6......$
So every I is closed. so $\bigcup(I) = (1,0].$
As this is open so the union of closed sets do not have to be open always

[pulsar trail]

Someone_Random

[scarlet pier]

i suggest writing $I_n$'' instead of $I$'' for clarity

your countable union shld b ``$\bigcup_{n \in \bN} I_n$''

$(1,0]$'' is wrong, it shld b $[0,1)$'', which is NOT open

[pulsar trail]

vin100

[scarlet pier]

you're close to what the question is asking

exercise: find a (countable) union of closed and bounded intervals such that it is an open (and bounded) interval