Showing that $\lim\limits_{n \to\infty} z_n = A$ implies $\lim\limits_{n \to\infty} \frac{1}{n} (z_1 + z_2 + \ldots + z_n) = A$
On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $
How to deal with "java.lang.OutOfMemoryError: Java heap space" error? At the end of The Return of the King, Frodo, Bilbo, Gandalf, Elrond, and Galadriel left Middle-earth on a ship, along with other elves. I named those five because they were all ringbearers and it seems to be a common trait. Frodo and Bilbo bore the One Ring and the three others bore the three elven Rings.  Why did they have to do that? Why didn't they stay and have a peaceful life now that the Dark Lord was gone? Double calculation producing odd result How does sudo handle $HOME differently since 19.10? I just stumbled upon a :     My wife and I's seafood collaboration dinner. How does it look?   Sure enough, the top comment immediately points out that it should be "my wife's and my". However, a cross-post to the Grammar subreddit produced :     It's fine as it is written. "my wife and I" is a noun phrase, functioning as a subjective pronoun in the singular and made possessive with the apostrophe. It is exactly the same as "our".      It seems weird because you would never use "I's" on its own but it is not on its own here - it is part of a noun phrase.   That's a rather intriguing argument. Does it hold any water? If a creature with Multiattack is affected by Haste, can it make two Multiattacks? It's a fairly straightforward question.   A creature has a Multiattack option, and it also can cast the Haste spell. While under the effect of Haste, can it use the second Action to take another Multiattack Action?  My main concern is the clause of the Haste spell that limits the number of attacks made from taking the Attack Action. This is a different Action, though.   Then there is the concern of being able to use Multiattack at all, because Haste says what kinds of Actions you can take. How do I install a .deb file via the command line? How do I install a .deb file via the command line? I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.   $$\frac 1{x+1}&lt;\ln\left(\frac{x+1}x\right)&lt;0.5\left(\frac 1x+\frac 1{x+1}\right)$$ Which one is correct? (Maybe both are correct.)     He passed away on the morning of March 5.   Or      He passed away in the morning of March 5. Does magnetic monopole violate $U(1)$ gauge symmetry? Does a magnetic monopole violate $U(1)$ gauge symmetry? In what sense and why? Insofar as I know, there are at least two types of magnetic monopoles. One is the Dirac monopole while the other is the monopole in Grand Unified Theories (GUT), e.g., 't Hooft-Polyakov monopole.  In the latter case, some non-Abelian gauge theory breaks down to a (compact) $U(1)$ gauge theory wherein monopoles can be observed from a large distance. The monopole field is not from the $U(1)$ degrees of freedom and nothing is singular. On the contrary, Dirac monopole is singular (topological defect?) and we must introduce Dirac string or patches of vector potential $\vec{A}(x)$. Are these no more than artifacts arising when one tries to describe the monopole field within the $U(1)$ gauge theory by assuming $\vec{B}=\nabla\times\vec{A}$? It seems true since two Maxwell's equations that magnetic monopole might affect can be cast in the differential form $\mathrm{d}F=0$ and $F=\mathrm{d}A$ in Minkowski spacetime, from which $U(1)$ gauge invariance manifests for $\mathrm{d}(A+\mathrm{d}\chi)=\mathrm{d}A$.  However, I later noticed two seminal papers on Dirac monopole  &amp; , in which the authors claimed the following   electromagnetism without monopole $\rightarrow$ connection on a trivial $U(1)$ bundle electromagnetism with monopole $\rightarrow$ connection on a nontrivial $U(1)$ bundle And the wave function of an electron around a Dirac monopole should be regarded as a section which is free from discontinuities.   I've no idea about fibre bundle. Anyway I guess $U(1)$ theory does not necessarily expel Dirac monopoles. It has something to do with the topology of some manifold in the theory? Which manifold? Can anyone shed light on this intriguing issue? Thanks in advance. How do I diagnose the source of focus problem in a camera? Probably a novice question, but how do I correctly diagnose from a photo for what reason that camera wasn't able to focus properly?  It can be any 1 of the following   Unsteady hand Lens issue Metering Re-Calibration required in camera or ...   Is there a set of unsaid protocols that needs to be followed by a photographer?  Goal is to get a sharp and crisp image prior to editing.  [PS: I own a Nikon D5200.] nmake fatal error U1034: syntax error : separator missing Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that there exists a simple closed curve, which is not null-homotopic in $U$.  I am sure that this is a standard problem, and I am looking for an elegant, if possible, solution or a reference. I have in mind the following sketch of proof:  a. Construct a polygonal closed curve $\tilde\gamma$ which is homotopic with $\gamma$ in $U$.  b. Construct $\tilde\gamma$ so that it intersects itself in finitely many points, and not whole segments.  c. Use induction on the number of self–intersections of $\tilde\gamma$, in order to show that $\tilde\gamma$ is written as a product (in the fundamental group's sense) of simple closed polygonal curves. How to delete a file from Time Machine backups in Big Sur? it VS. this Vs. that How can I backup my RAW photos while travelling without Internet access? This question seems similar to , but my needs are different :   I shoot in RAW exclusively I'm looking for a 100% offline solution: while traveling, Internet won't be accessible the "laptop" solution is too heavy for me: I'm looking for a lightweight solution the simplest, the better: one device to rule them all   EDIT:  I'm using good old Compact Flash cards (yes, the bigger ones), so I need a device that can read it. If it can read from other cards, this question might be interesting for more people than myself  - that's the whole purpose of the website.    Also asked by :  Portable Storage Device while traveling  I need storage device which is able to read CF cards and make a backup of my photos while traveling. I do not want to carry laptop (it's bulky) and tablet (too little storage) but looking into some other solutions.   So far I found:   Epson PXXXX - price is blocker for me Jobo Giga Vu - same as above Nexto DI - bit better, but still a lot Hyperdrive Colorspace UDMA - this looks the best for price/options Wolwerine PicPac - cheap, bad opinions DigitalFoci PhotoSafe - same as above Ex-Pro® Picture 2 Drive - didn't find much about it   So ladys &amp; gents what do you use? We now that Fermat's last theorem is true so there are not positive integer solutions to $$x^n+y^n=z^n$$ for $n\in\mathbb{N}$ and $n&gt;2$.  But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$? Typeset two styles of the same text over top each other Given that the sequence $(v,Av,A^2v,...,A^nv)$ is linearly dependent (1), $A_{n \times n}$, over $\mathbb{F^n}$ prove that scalars $c_0,c_1,..,c_n$ exist (such that not all of them are zeroes) such that the matrix $M=c_0I+c_1A+..+c_nA^n$ is singular.    First I'm not sure how (*) can be used here exactly. We can say that the sequence $(c_0I,c_1A,..,c_nA^n)$ resembles the sequence (1) where $v=I$. But here we have scalar $c_i$ so it's not exactly the same sequence.   Maybe I could say that the sequence $(c_0I,c_1A,..,c_nA^n)$ is linearly dependent because it has $n+1$ elements while over $dim\mathbb{F^n}=n$ $\quad$(2).  Intuitively I understand that suppose we sum all the matrices $c_iA^i$ and then in the last sum operation we get to some $c_jA^j$ and if it's a linear combination of other matrices then we can have such a $c_j$ that: $$ c_oI+\sum_{i=1}^{n-1} c_iA^i+c_jA^j=0 $$ But I don't think this is enough to prove. Does Two-Face die at the end of The Dark Knight?