I can't put into words what part b is suggesting
#Real analysis
half prawn
brisk ginkgoBOT
I can't put into words what part b is suggesting
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half prawn
like u remove a_n from the sequence
if a_n was the infimum, then the infimum increases (dk how to show this)
if it wasn't the infimum, then the infimum stays the same (also dk how to show this)
nocturne inlet
I can't put into words what part b is suggesting
i'll call the (``beheaded''-)subsequence $$a_n, a_{n+1}, \dots$$ an \emph{$n$-tail} of $\qty(a_n)_n$.
dusty pondBOT
vin100
nocturne inlet
that's just "liminf limsup". it's not hard to find pic like
the result is quite visually intuitive
half prawn
yeah i can see why b_n and c_n converge to the limit of a_n
but i dont know how to put it into words
nocturne inlet
but i dont know how to put it into words
$\qty(a_n)_n$ has ``diminishing tails''.
dusty pondBOT
vin100
nocturne inlet
using the above graph, the blue dots' fluctuations are "eventually small"
$\qty(b_n)_n$ (n-tails' lower bound) and $\qty(c_n)$ (i.e. n-tails' upper bound) will converge, since the tail is getting "smaller and smaller"
dusty pondBOT
vin100
nocturne inlet
ℹ️ i'm putting some words in "double quotes" just to give you verbal intuition, since you've asked to put things in words. they are intentionally vauge, and they aren't rigorously defined
half prawn
by words i meant maths words lol
sorry
i know the intuition, im struggling to formalise it
b_n is monotonically increasing and i feel like its bounded above by the limit of a_n
c_n decreasing and bounded below by the same thing
but to say its bounded is not enough to know its limit ?
just to know it converges
and also, idk how to show b and c are bounded by this limit of a
nocturne inlet
but to say its bounded is not enough to know its limit ?
the question has given what you needed
,,b_m \le c_n
dusty pondBOT
vin100
nocturne inlet
i'm being a bit lazy not to write the quantifiers and domain associated with m and n
from this you can use MCT
nocturne inlet
**vin100**
use capture this intuition in rigorous math
a simple math word would be "Cauchy"
but i think you might not need Cauchy here
you're alr given a convergent sequence
so just use the epsilon-N definition then take supremum
after unwrapping the absolute sign
half prawn
i still dont get what you mean sorry
nocturne inlet
i still dont get what you mean sorry
you don't know how to use this in your question?
half prawn
yeah
as in
b is bounded above sure by some c_n
and its monotonically increasing so by mct, it converges
then wdym by take the supremum
how can we do this
nocturne inlet
so you have $b_n \uparrow b$ and $c_n \downarrow c$
dusty pondBOT
vin100
nocturne inlet
b_n is increasing and converges to b?
when you get stuck shifting from one sense to another sense might give you some inspiration
half prawn
c and b eventually become equal to the limit, i understand
nocturne inlet
that's the complete idea (with poor cAsE and punctutations)
nocturne inlet
you're asked to prove a proposition the starts with "for all", and the standard first step is "let/fix ...."
i didn't say "take sup of b"
,tex nor ``take sup of $\qty(b_n)_n$"
dusty pondBOT
vin100
half prawn
what does take sup mean
nocturne inlet
what does take sup mean
take supremum
nocturne inlet
what are u taking the supremum of
write out the epsilon-N definition first
then you might know what to take
half prawn
for all eps, theres an N st for all n > N, |b_n - b| < eps
nocturne inlet
for all eps, theres an N st for all n > N, |b_n - b| < eps
you're given $\qty(a_n)_n$ convergent, so you can use this
dusty pondBOT
vin100
nocturne inlet
but i dont see how the definitions help
you haven't even written it down
half prawn
for all eps, theres an N st for all n > N, |b_n - b| < eps
its just the same as this but replacing the b with a
is it not
nocturne inlet
yeah then i've said "to unwrap the absolute sign"
using words, the n-tails is eventually contained inside an epsilon-neighbourhood of a
i.e. $(a-\epsilon, a + \epsilon) \subseteq \bR$
dusty pondBOT
vin100
nocturne inlet
$b_n$ is just the greatest lower bound of this tail, and similarly for $c_n$
dusty pondBOT
vin100
nocturne inlet
the closed interval $[b_n, c_n]$ is like a tight clothing wrapping the $n$-tail ${a_k \mid k \ge n}$
dusty pondBOT
vin100
nocturne inlet
i.e. ${a_k \mid k \ge n} \subseteq [b_n, c_n]$
dusty pondBOT
vin100
nocturne inlet
so we should expect this tight clothing inside the epsilon neighbourhood
the choice of epsilon is arbitrary, so ...
mellow ingot
@half prawn if the sequence does converge to some limit L, it's eventually arbitrarily close to it. Now suppose we have, for all m>n
L-e < x_m < L-e
Also if we look at the set, {x_i : i >= m} and it's infimum can we say that,
L-e <= inf{x_i : i >= m} <= x_m?
half prawn
<@931275367426179102> if the sequence does converge to some limit L, it's eventu...
hmm yeah
mellow ingot
if we can can say that why?
why can we say that?
after you figure it out do the same for the supremum
half prawn
alright thank you
mellow ingot
if you'd like the proof fully written out lemme know btw
i suggest you to try it yourself of course
half prawn
yeah I'll attempt it rn
mellow ingot
aight I'll be back in a few mins
half prawn
L-e is a lower bound for the x_i and the infimum is the greatest lower bound so its by definition greater than L-e?
mellow ingot
yes
half prawn
then can we apply thr sandwich lemma?
mellow ingot
bro if you call it sandwich I'll beat you up
half prawn
oh wait
mellow ingot
half prawn
😂
mellow ingot
sandwiches need not be squeezed
half prawn
we already have L-e < inf(x_i) < L+e
mellow ingot
squeeze OUGHT TO BE SWUEEZEF6