This is part of an old preliminary exam in Analysis I am reviewing to prepare for my own prelim. $\lambda$ is the Lebesgue measure. $f_n\to f$ with respect to the $L^1-$norm.  I know that there exists at most a subsequence fulfilling the condition, since, e.g., we could have $f=0$ and $f_n=0$ except at $q_n$, with $q_n$ cycling through the rational numbers (with each rational number appearing infinitely often). Then $f_n\to f$ but it is not the case that $f_n(x)\to f(x)$ for almost every $x\in[0,1]$. It's obvious in this example that there will exist a subsequence fulfilling the condition, and I feel confident that there will always be such a subsequence. That is, I think the answer is to "prove", not "disprove". But I'm having difficulty seeing how to prove it.  This is a case where I feel like I understand the problem but can't quite figure out how to structure a proof. Every attempt I've made has become overly convoluted and I've become confused. Any help would be appreciated.
I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are  and my particular question is from Section 4, Corollary 3. It goes as follows,  Let $f_n \rightarrow f$ in $L^1$ then there exists a sub sequence $(f_{n_j}) \subset (f_n)$ such that $f_{n_j} \rightarrow f$ pointwise a.e. Moreover $(f_{n_j})$ converges almost uniformly to $f$.  The proof he gives is simply that since $\|f_n-f\|_1 \rightarrow 0$ as $n \rightarrow \infty$ we can pick a sub sequence such that $\|f_{n_j}-f\|_1&lt;2^{-j}$ which is enough to show pointwise a.e and almost uniform convergence. But what allows you to pick such a sub sequence is it maybe some Cauchy property or is it some weird construction? Then how do you go from that to pointwise a.e and even almost uniform convergence. I am assuming that for almost uniform, you do something similar to Egorov's theorem without the assumption the domain of $f$ has finite measure. Also I am aware that if you get almost uniform, you immediately have pointwise a.e, but I'd like to see how to get to both. Thank you.
Time traveler asks archaeologist for help in transporting a girl from dinosaur Earth to present day Many decades ago, I read a short story about a time traveler who asks an archaeologist for help.  He convinces the archaeologist that he can time travel by bringing him back a freshly killed dinosaur (small one.)  The time traveler needs help because when traveling back in time to the age of the dinosaurs he met a girl and wants to transport her back to the present.  However only one person can travel in time at a time so he needs the archaeologist at the other end to wait for the girl.  The archaeologist agrees to help.  Unfortunately after the traveler goes back he and the girl find themselves in the middle of a battle between aliens on Earth.  They are chased by a dinosaur and never make it back to the present.  The archaeologist meanwhile is digging for evidence that the traveler went back in time and the story or my recollection of it ends with that. Yet another "inconsistent reputation history" Use of 'const' for function parameters Why does inductor current lag the applied voltage at its terminal by 90 degrees? I studied electromagnetism and I am currently working with Inductors.  I could not figure out the physical reason based on electromagnetics on why inductor current lags the applied voltage by 90 degrees?  Everywhere I find the Formula as the starting point of explanation which is partially satisfying.    Induced EMF =L di(t)/dt    From the above equation, I understood that when there is a change in current there will EMF induced in the coil terminals. But How this induced EMF makes the current to lag the supply voltage V(t) = Vm*sin(wt)? (Please don't just differentiate and say there is a lag, I am looking for a physical sense on why this happens)  Lets us consider the AC voltage is applied across the terminal at time t=0; At this time there is almost no current. From this transient state how the steady-state is reached where the current is delayed with respect to the voltage?  Can someone help me to figure out why the current lags the Supplied voltage by using electromagnetism, the motion of Charges or any other basic concepts? Is a comma necessary in "What’s funny, is …"? In the sentence     What’s funny is Cat’s dry humour.   Wouldn’t it be better to put a comma after “funny”:     What’s funny, is Cat’s dry humour.   This question entails a (at the time of writing) heated discussion in the . I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated. I'm answering this question:     Let $M$  be a module. Show that if  $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever $N &lt; A ≤ M$.   The solution starts as follows (I'm following a given solution): Let $Y$ be the set of non-finitely-generated submodules of $M$. If $(N_α)$ is a chain in $Y$ then $Q= \bigcup_αN_α $ is in $Y$, else $Q$ is f.g., which implies $Q=N_α$ for some $α$, whence $N_α$  is f.g., contradiction.  I don't understand why if $Q$ is f.g., then $Q=N_α$ for some $α$? Am I missing some fact?  Thanks! The following exercise comes from Fulton's Algebraic Curves:     Let $k$ be an infinite field and $V = \{(t,t^2,t^3) \, ; \, t \in k \} \subset \mathbb{A}^3(k)$. Show that $V$ is algebraic and compute $I(V)$ by giving a finite set of generators.   Now, showing $V$ is algebraic is easy by noting that $V = V(X^2-Y, X^3-Z)$, but I can't find a way to describe $I(V)$ explicitly. Obviously $(X^2-Y)(X^3-Z) \in V$, but how can I find the actual generators of this ideal? Independent Edit/Object mode in splitscreen Is there an easy way to attach source in Eclipse? I'm a big fan of the way Visual Studio will give you the comment documentation / parameter names when completing code that you have written and ALSO code that you are referencing (various libraries/assemblies).  Is there an easy way to get inline javadoc/parameter names in Eclipse when doing code complete or hovering over methods? Via plugin? Via some setting? It's extremely annoying to use a lot of libraries (as happens often in Java) and then have to go to the website or local javadoc location to lookup information when you have it in the source jars right there! Why does Python's itertools.cycle need to create a copy of the iterable? How to automatically start programs in specific workspaces on start up Within a custom command, I'm doing the following to write to a file:  \newcommand\mycommand[1]{   \newwrite\file   \immediate\openout\file=foobar.tex   \immediate\write\file{     foo     bar     #1     blarg   }   \immediate\closeout\file }   Of course when I invoke it as \mycommand{baz}, I get the following contents in foobar.tex:  foo bar baz blarg   with all new lines turned into spaces by LaTeX. My question is, is there a way to output these lines into the file "as expected", i.e. with real new lines at the end of each line instead of a space? I'm thinking some \catcode wizardry should do the trick, but I'm not sure which character I need to redefine and to what category code.  Update  The solution in  works perfectly. The only problem is when #1 contains embedded newlines. I can of course send it to mycommand with its own slew of ^^J, I was wondering if there is a way to avoid this. Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$ Theorem: There are infinitely many primes congruent to $3 \mod 4$. Proof: Assume that $p_1 = 3, . . . , p_n$ are primes of the form $p_j ≡ 3 \mod 4$. We will construct a new one by looking at $N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work).  There is more to this proof but the rest of it I understand, what I don't get is where it says &quot;$N = 4p_1 · · · p_n − 1$ (putting $N = 4p_1 · · · p_n+3$ would also work)&quot;. I do not understand why we are writing $4p_1 · · · p_n − 1$. Is there a more clear way to prove this theorem? Beamer - create subsubsection page It is asked to solve the PDE  $$u_x + u_y + u = e^{x+2y}$$  My attempt: We have that $$\frac{dx}{1}=\frac{dy}{1}=\frac{du}{e^{x+2y}-u} $$  $\Rightarrow \frac{dy}{dx}=1 \Rightarrow y=x+C_1 \iff C_1=y-x$  $\Rightarrow \frac{du}{dx}= e^{x+2y}-u \iff \dot{u}+u=e^{x+2y} \iff \\ \frac{d}{dx}(e^x u) = e^{2x+2y} \iff u = e^{-x} ( \frac{1}{2}e^{2x+2y} + C_2)$  As $u$ is constant along its characteristics:  $$u(x,y)=\frac{e^{-x}}{2}(e^{2x+2y} + 2C_2(y-x))$$  where $C_2$ is any function of one variable. But wolfram gave the following answer  $$u(x,y)=\frac{e^{-x}}{4}(e^{2x+2y} + 4C_2(y-x))$$  as you can check   What am I doing wrong? Are there any drawbacks to legacy BIOS as opposed to UEFI? Are there any drawbacks to installing Ubuntu 13.10 (and soon 14.04) in Legacy BIOS instead of UEFI?  I have installed 13.10 64bit in Legacy by mistake and was just wondering if it would hurt anything to simply leave it there as is?  EDIT  Everything seems to be running just fine in Legacy and all my drivers work. Find the value of $\space\large i^{i^i}$? Is $\large i^{i^i}$ real ? How to find it?   Thank You! Are there any disadvantages to always using nvarchar(MAX)? Why is "No new release found" when upgrading from a LTS to the next?