Show that if $(a_k)_{k \in \mathbb{N}}, (b_k)_{k \in \mathbb{N}}$ are bounded, then $\lim \sup(a_k + b_k) \leq \lim \sup(a_k)+\lim \sup (b_k)$  I'm not quite sure how to prove this.  I want to try this comparing cases.  Assume that $a_k \geq 0 \forall k \in \mathbb{N}, b_k \geq 0 \forall k \in \mathbb{N}$  $\Rightarrow \sup (\bigcup_{i=k}^\infty \{a_k + b_k \}) = \sup (\bigcup_{i=k}^\infty \{a_k \}) + \sup (\bigcup_{i=k}^\infty \{a_k \})$   The same is true if both sequences are negative for all k.  If one of the sequences is negative for all indices and the other is positive, then the inequality would be real.  Would that be sufficient for a proof, or does the fact that e.g. all terms are positive in both sequences not imply the equality of the sup terms?
I am stuck with the following problem.     Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$   I was thinking of using the triangle inequality saying $$|a_n + b_n| \le |a_n| + |b_n|$$ but the problem is not about absolute values of the sequence.  Intuitively it's clear that this is true because $a_n$ and $b_n$ can "reduce each others magnitude" if they have opposite signs, but I cannot express that algebraically...  Can someone help me out ?
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere? I wanna prove that   "if $f: \mathbb{R}^n \to \mathbb{R}$ is continuous and satisfies $f=0$ almost everywhere (in the sense of Lebesgue measure), then, $f=0$ everywhere."  I am confident that the statement is true, but stuck with the proof. Also, is the statement true if the domain $\mathbb{R}^n$ is restricted to $\Omega \subseteq \mathbb{R}^n$ that contains a neighborhood of the origin "$0$"? Downgrade nVidia Shield to Android 4.4 KitKat How to prove that a simple graph having 11 or more vertices or its complement is not planar? It is an exercise on a book again. If a simple graph $G$ has 11 or more vertices,then either G or its complement  $\overline { G } $ is not planar.  How to begin with this? Induction?  Thanks for your help! Set active image node with python There's any way to change it directly by Python, or there's any way to set by Python what node image is the active to display in viewport (Textured) and bake? Remove empty elements from an array in Javascript Para 5, , by      If the Lord would show thee but this one   thing, -- that to use "thee" and "thou" to a particular person is   proper language, and Scripture language; and that to say "you," is   improper, and arose from pride, and nourisheth pride, and so is of the   world, and not of the Father; and thou should bow thy spirit to him in   this one thing, thou little thinkest what a work it would make within thee, and how strongly the spirit of darkness would fight against thy   subjection thereto. The Lord lead thee as he seeth good, and give thee   faithfully to follow; for else, if the Lord should lead in any thing,   and thou not follow in that thing, his Spirit would be grieved and   vexed thereby, and thy heart in danger of being hardened by the   deceitfulness of sin.   What is the Lord giving thee faithfully to follow?   The grey claims that the reader thinks little of the significance of stressing 'thee' and 'thou' over 'you'. So I'd guess that the Lord is giving 'thee' this caution? If I'm right, why is there no direct object in the bolded? I recognise grammatical differences in the 1600s, but still want to learn. In the following figure prove that: $AK||BC$ In arbitrary triangle $ABC$, let $G$ be centroid and $AH$ be a height.We extend $HG$ beyond $G$ to intersect circumcircle at $K$.Prove that $AK||BC$       I connected midpoint of $BC$ to $K$ and $A$ to show $MA=MK$ but failed... The problem from Burton: show that the equation $n^2 + (n+1)^2 = m^3$ has no solution in the positive integers.  So far, I can see that gcd($n$,$n+1$)$=1$ and $m \equiv_4 1$ and $m=a^2 + b^2$ for some integers a,b. I'm guessing I need to reach a contradiction.  At this point, I am stuck. Any hints? How can i show with quite simple methods that a group of order 15 is cyclic? Making overview with linked detailed maps in QGIS? i want to create 4-5 detailed maps, and then an overview map that has rectangles that markes the locations of the detailed maps. Best case would be if the rectangles were links to the detailed map.   I have managed to make the overview and the rectangles, and think I know how to make the detailed maps one by one. The rectangles are drawn manually on the overview map. All layers that I plan on using are in the same QGIS project.  The final product that I hope to create would be the overview map on the first page, followed by the detailed maps. The overview would have links to the detailed maps, click on the rectangles to open the right detailed map.  Is there a way to link the detailed maps to the overview in QGIS 2.8 print composer?  I'm still not sure if I'm doing something wrong, when I try to create the overview. I want it to show the extents of all of my detailed maps.   Is this possible? Getting computer's UTC offset in Python Question mark or bold citation key instead of citation number I've browsed the forums and found a number of posts that have addressed this issue, but none of the solutions seem to work for me. I have the following script that I just copied from the bibtex home page to get familiar with it. Instead of the citation number I get a question mark. I compile using Latex+Bibtex+Latex+Latex+PDFLatex+ViewPDF just as has been previously suggested and the problem persists.   \documentclass[11pt]{article} \usepackage{cite}  \begin{document}  \title{My Article} \author{Nobody Jr.} \date{Today} \maketitle  Blablabla said Nobody ~\cite{Nobody06}.  \bibliography{mybib} \bibliographystyle{plain} \end{document}   My bibliography (Bib.bbl)  @misc{ Nobody06, author = "Nobody Jr", title = "My Article", year = "2006" }    Looking at previous posts one thing that is concerning is that my .bbl looks empty as shown below. Further, I don't have a .blg  \begin{thebibliography}{}  \end{thebibliography} Is it appropriate to buy a "thank you" gift for a PhD supervisor? I am nearing the end of my PhD and I would quite like to buy a "thank you" gift for my supervisor (although, she's the socially awkward type who I'm sure will be greatly embarrassed to receive one).  I'm just wondering whether a gift is the done thing? And if so, what sort of gift seems appropriate? I suppose a bottle of wine would be a safe choice? I am having trouble with (a) and (b):  I evaluated (a) and I got that sun of the first three terms does not equal $\ln(5/8)$. I got that it is equal to $\ln(3/4) + \ln(2/3) + \ln(15/16)$ which is not equal to $\ln(5/8)$.  I am also having trouble with part (b). I cannot figure out how to calculate the math to cancel out the $(k+1)^2$ and turn it into $2(n  +1)$.  Any help would be amazing. Trouble plotting arctan with pgfplots I'm trying to plot arctan(x) with pgfplots but I don't get a good output. Here is my code:   \begin{cfigur}\begin{axis}[      width=160pt,compat=1.5.1,grid style={ultra thin},every axis plot post/.append style={thick},      x tick label style={font=\tiny},y tick label style={font=\tiny},      scale only axis,grid=major,axis lines=middle,      xlabel={$x$},      ylabel={$y$},      xmin=-200,      xmax=200,      domain=-200:210,      ymin=-5.5,      ymax=5.5,      xtick={-150,-100,...,150},      ytick={-5, -4,...,5},      restrict y to domain=-20:20,      legend style={at={(0.5,-0.05)},anchor=north,nodes={right}},  ]  \addplot[mark=none,color=blue, samples=500]{atan(x)};  \addlegendentry{$y = \tan^{-1}x $};  \end{axis}\end{cfigur} USB 2.0 device (scanner) does not work with xhci_hcd on USB 3.0 system Sakurai QM section 5.8 If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He starts with (3) (given below), and derives (1) and (2). The equation is as follows:$$(1){\space}{\space}\space{\space}\space{\space}{\dot c_i\over c_i}\approx{{-i\over \hbar}V_{ii}+{\left(-i \over \hbar\right)}^2{{{\left|V_{ii}\right|}^2}\over \eta}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}} \over 1-{i \over \hbar}{V_{ii} \over \eta}} $$ $$(2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\approx{-i\over \hbar}V_{ii}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}}$$ where $$(3)\,\,\,\,\,\,\,\,\,c_i(t)=\lim_{\eta\to 0}\left({1-{i\over \hbar\eta}V_{ii}}{e^{{\eta}t}}+{{\left({-i\over \hbar}\right)}^2}{{\left|V_{ii}\right|}^2}{e^{2{\eta}t}\over 2{\eta}^2}+\left({-i\over \hbar}\right){\sum_{m\not=i}{{{{\left|V_{mi}\right|}^2}}\over 2\eta(E_i-E_m+i{\hbar}\eta}}\right)$$First of all, he discards all kinds of terms in the denominator of the first expression, and I can't see how he justifies this, because as ${\eta}\to0$, every term in the expression for $c_i(t)$ blows up (besides the 1 of course). Then he seems to multiply top and bottom by the complex conjugate of the denominator in (1), arriving at (2). He seems to discard a term from the denominator here, which is of second order in V, and says the equation is formally correct up to second order. This doesn't make sense to me! Also, soon after, he has this strange expression with $$\lim_{x\to 0} {1\over(x+iε)}=Pr.(1/x)-iπδ(x)$$ What is this Pr symbol? Any help would be much appreciated! Does %* in batch file mean all command line arguments? 2015 13" MacBook Pro Trackpad and Keyboard stop working I'm from the State of Kuwait and hold an Eritrean passport. I'm planning to visit UK in October but my visa will expire in November.    Is it OK to make a journey in the last month of my visa, provided that I make sure that I book my plane back before expiring date?