it's like how points are labeled in elementary geometry. people often start with A, B, C, etc. but a labeling of the points on a triangle isn't generally an attempt to import the ordering of the alphabet on points in a triangle
it's sometimes a guide to the order in which you draw or construct things in the course of an argument, but it's not of mathematical significance
that bothered me a lot when i first learned geometry. draw triangle ABC. then you realize that whatever you've drawn, you have lots of choices in how to label it with A, B, and C
in cultures that write left-to-right i think it's common to label A, B, C left to right to the extent possible, but that's not ordering by x-coordinate or anything either
it's just, I'm used to thinking of a set $\{ i : i \in I\}$ as sort of a mechanism that gathers up each $i \in I$ and fills up an empty set, one by one. i'm trying to undo that way of thinking about it since it implies an order
we all learn this without realizing that we learned it until someone hits us with a set indexed by something like "the set of all all open intervals in R"
finite group theory might be another example where you sometimes see indexing that doesn't matter. people will sometimes say ok, the group G is finite, so write G = {g_1, g_2, ..., g_n} with the g_i distinct, just so they have some notation for referring to different elements of G
unless the group is cyclic and you label according to that, or label according to some other specific scheme adapted to the group law, the indices won't have any relationship with the group operation
g_1 * g_2 isn't going to be g_{1+2} or anything like that
Let $f_n(x): \mathbb{R}\to\mathbb{R}$ be a sequence of measurable functions.
Show that the set $$\{x: (f_n(x))_{n=1}^{\infty} \ \text{converges to a real number}\}$$
is measurable.
My attempt:
$\begin{align}
\{x: (f_n(x))_{n=1}^{\infty} \, \text{converges to a real number}\}&=\{x: (f_n(x))_{n=1...
it's translating the definition of what it means for a sequence to be cauchy, into set operations that are known to preserve measurability
1/k is playing the role of "epsilon" here; they're using the sequence 1/k because they need a countable intersection
the fundamentals of measure theory are full of proofs that you can do this way, although the unions and intersections you write down are not things you would generally consider if you weren't tuning an argument directly at the operations that preserve measurability
the limit exists at x iff for all k there is N such that for all n, m >= N, |f_n(x) - f_m(x)| < 1/k. it's that, rewritten in set operations
@robjohn about that alternating sum from last night. i don't know how to evaluate it but per computers it's a multiple of zeta(1/2), which is apparently the limit of (sum_n=1..k 1/sqrt(n)) - 2 sqrt(k), which connects it right back to one of my favorite divergent sums
I have posted a number of answers using the Euler-Maclaurin Sum Formula to analytically extend things like the divergent sums minus their parts that don't converge.
my poor wife is trying to have a work meeting in another room and has shut a door. this is rare, normally we just have our work calls wherever but i think they are discussing sensitive stuff. it is driving the cat crazy to hear talking in a room that she isn't allowed into.
twice now i've had to pick her up to keep her from thumping the door with her paws and meowing
@Mast I just checked, it seems to be. However, I am more than a little bothered by such accusations. I am comfortable with some back & forth and am generally tolerant (if a little hot headed at times) but that crossed the line.
I understand. Please, try not to let the hot headedness flow into the conversation. If a message crosses the line, keep calm and flag it. All will be sorted out, there's always around.
Let $\displaystyle f:[ a,b]\rightarrow \mathbb{R}$ be a continuous function, that is $\displaystyle f\in \mathcal{C}[ a,b]$. It is to be proven that $\displaystyle m\in \mathcal{C}[ a,b]$, where $\displaystyle m( x) =\min_{a\leq t\leq x} \ f( t)$.
@Koro I am not a fan of sequence proofs, but I think for continuity of $\max,\min$ functions, it is easier to use sequences. I am still reading your proof, I am slow.
hard to say. i would look at a lot of examples (if your book doesn't include them, find others). constructions involving step functions, spikes that get tall as they get narrow, spikes that move around, etc. are fairly common in the area of counterexamples, but due to the number of arbitrary choices involved, i don't know that there's a specific route through these techniques.
in grad school one time we needed a team name for a quiz, and we called ourselves The Roving Spike after a counterexample that was something like, you enumerate the rationals on R, and put a spike of some height and width centered on the nth rational. i forget what it showed but i remember our name for it.
the roving spike would also be a good name for a pub.
various theorems (e.g. MCT, DCT) and their proofs sometimes shed light on how bad a counterexample will have to be. a lot of counterexample ideas come out of that, but there is still usually a wide universe of choices to be made in picking one
one vibe is that a lot of integration examples involve pushing mass out to infinity, or up to infinity (i.e. very high but narrow peaks)
and a lot of hypotheses on measure spaces (e.g. finiteness of the measure space) or functions (e.g. various integrability conditions) put you in a box where you can't do that and everything works
@geocalc33 Well, the natural log is an isomorphism from the multiplicative structure on $\mathbb{R}^+$ to the additive structure on $\mathbb{R}$, so you could probably say that it's just a lattice
I havent used set notation before, and when I was trying to differentiate the different taylor series making up e^x I had all these long winded expressions which were my way of saying what I wrote above
@robjohn that was just an example, of the sort of thing I was wanting
if I have a differential equation of the form $$a \frac{\partial }{\partial y} f(x,y) = b \frac{\partial ^2}{\partial x^2} f(x,y) + g(x,y)$$ what would it mean to find a function $g(x,y)$ that is "a solution to this equation"?
sorry the differential equation is schrodingers: which I was just digging up, it is : $$ \frac{\partial \Psi}{\partial t} = \frac{i \hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar} V \Psi, $$
(i was replacing psi with f and v with g)
okay, so in my case, f is a given. but g is unknown. But you say it is uniquely determined, so if I evaluate the differential equation I get a unique expression that 'solves' the equation?
hopefully when you do that, the formula you get for V has whatever properties it needs to have to be a 'potential'
it's asking for the function V for which a specific Psi satisfies that equation
any symbolic expression can have any number of 'solutions' depending on what you regard as given and what you regard as unknown. i think it's common for V to be given and that equation to be solved for Psi, so maybe that's why they're talking about, "for which V is Psi a solution to ..." instead of referring to V as a solution to that equation
but the mechanics of finding V would just be algebra, if i'm understanding this
so I'm right to think a "solution" is just some expression $V(x,y) = ...$ where $...$ may contain $\Psi$ and it's derivatives?
ie (where I expand $\Psi$ appropriatley from the equation given above): $$ V = \frac{2m\hbar^2}{\Psi} \frac{\partial^2 \Psi}{\partial x^2} + \frac{i}{\Psi}\frac{\partial \Psi}{\partial t} $$ assuming I haven't made any stupid algebra errors
