I can't seem to figure out why the proof is false. Everyone knows that $1 \ne 2$ and that $1=2$ is only proven through mathematical fallacies.  However I can't figure out where the problem is in the following fallacy:    Observe that  $$1=\frac{2}{3-1}$$  Substitute $\frac{2}{3-1}$ for the $1$ in the denominator  $$1=\frac{2}{3-\frac{2}{3-1}}$$  Substitute again $$1=\frac{2}{3-\frac{2}{3-\frac{2}{3-1}}}$$  Therefore $1$ can be written as the infinite fraction: $$1=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots}}}$$  Now notice that $$2=\frac{2}{3-2}$$  Substitute 2 similar to how 1 was substituted above  $$2=\frac{2}{3-\frac{2}{3-2}}$$  And through repeated substitution we get $$2=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots}}}$$  Notice that $1=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots}}}$ and $2=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots}}}$  Therefore, $1=2$    Just looking at this I know that it cannot be true. But they both seem to share the same infinite fraction, so what is it that I am missing?
It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$  Now,   $$1=\frac{1}{2-1}=\frac{1}{2-\cfrac{1}{2-1}}=\frac{1}{2-\cfrac{1}{2-\cfrac{1}{2-1}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-1}}}}=\ldots =\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\dots}}}}},$$  $$2=\cfrac{1}{2-\cfrac{3}{2}}=\cfrac{1}{2-\cfrac{1}{2-\cfrac{4}{3}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{5}{4}}}}=\cfrac{1}{2-\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{6}{5}}}}}=\ldots =\cfrac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\frac{1}{2-\ldots}}}}}.$$  Since the right hand sides are the same, hence $1=2$.
Convince grep to output all lines, not just those with matches Does payment in goods count as "income" for tax purposes? Can someone help me elucidate the definition of "income" from a tax perspective? More specifically, if Joe is paid for his work in avocados instead of dollars, then would Joe be exempt from paying any income tax for the year? Or would the value of avocados earned for the year be translated into dollars which would then be considered Joe's annual income? If so, what guidelines would one use to make this translation? (Make the assumption that Joe does not sell the avocados (for dollars) at a later date and either eats them or trades them for other real property)  What if he was paid in gold instead of avocados? I have too many sections in my presentation such that the left-hand-side navigation bar is full and only shows the first sections in Fig. 2, initial situation in Fig. 1.  Pseudocode    clean the left-hand-hand side navigation bar when exceed the threshold and start a new one there when on sufficient page  this feature occurs with Beamer Berkeley appendix so it is possible also with the main document   Code which problem shown in Fig. 2   \documentclass{beamer} \usetheme{Berkeley}  \logo{\includegraphics{logo.png}} \usepackage{hyperref} % https://tex.stackexchange.com/a/346486/13173 \newenvironment{slide}[1] {\begin{frame}[environment=slide,allowframebreaks] \frametitle{\insertsection-#1} } {\end{frame}} % fragile only if code % allowframebreaks not by default   \begin{document}  \title{Sensory system. Primary sensory modalities. Cortical modalities. Kinds of sensory disturbances. Spinal cord disorders. Spinal cord transverse and partial lesion. Cauda equina damage.} \author{Leo}  \section{Cortical modalities} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 2} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 3} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \section{Cortical modalities 4} \begin{slide}{basics}      More refined aspects of sensation ... \end{slide}  \end{document}   Fig. 1 Situation at the first page,  Fig. 2 Situation at Section 6 wheere the navbar has not changed,  Fig. 3 Samcarter's answer output + one slide addition but without \part{} - fails on sufficient page count       Output: the left-hand-side navigation bar is not refreshed when the amount of sections exceed its capacity on suffcient enough page     Testing Samcarter's proposal  I include one slide before Samcarter's body so I get the wrong output in Fig. 3.  Solution: Eventually correct output by just adding \part{} in the place where the new navbar is wanted.   TeXLive: 2016 OS: Debian 8.7 I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? IBus methods do not show up in text entry settings I installed both Korean and Japanese using the Language Support window, and expected Korean (Hangul) and Japanese (Anthy) to show up in the input sources for Text Entry Settings (having both packages installed). However, neither shows up. Over the course of time here on TSE, I have observed individuals whose pattern of visa applications is best described as ''...     serial refusals - Two or more consecutive visa refusals issued by the same regime to the same person. Commonly for the same, or nearly the same, reasons. In some visa regimes Serial refusals can have a cumulative effect wherein the applicant's credibility is increasingly eroded, especially when the application pattern suggests a secondary agenda.   A common thread appears to be the need to visit the destination country (especially Schengen, the USA, or the UK) with a special sense of urgency. Also common in these cases are the following types of questions...   Should I just reapply once again? What are my chances? What can I do to improve my odds in the future? Can I somehow address the previous refusals to avoid them being a burden on all future applications? How can I break the pattern?      NB: This question is intended as a canonical about serial refusals. I have never been personally denied a visa. I would like to work out the Fourier transform of the Gaussian function   $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$  It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to work. Does anyone have any advice?  By the way, I am using   $$\mbox{FT}(f)(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,\mathrm d x$$  as my definition of a Fourier transform. Many thanks. Age and Graduate school A machine is described as being "really slow" or "freezing". How can I troubleshoot/resolve this issue? What should be the first steps to take? The direct sum $\oplus$ versus the cartesian product $\times$ In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times A$; however in, for example, the fundamental theorem of finitely generated abelian groups, we normally write that every such group is isomorphic to one of the form $$ \mathbb{Z}^n \oplus \mathbb{Z}_{r_1} \oplus \cdots \oplus \mathbb{Z}_{r_t}  $$ where $\mathbb{Z}^n$ now means $\mathbb{Z}\oplus\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}$.  From an intuition perspective, and in the sense of sets, is this more or less the same as $\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z}$? (Bear in mind I am normally using these ideas in relation to homology groups.) Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Embedding ordinals in $\mathbb{Q}$ Cannot locate Ubuntu Software Center via Firefox to open APT links Example 4.21 in Baby Rudin: Why is the inverse of this function not continuous at this point? Here's Theorem 4.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:      Suppose $f$ is a continuous 1-1 mapping of a compact metric space $X$ onto a metric space $Y$. Then the inverse mapping $f^{-1}$ defined on $Y$ by $$ f^{-1}( f(x) ) = x \ \ \ \ \ \ \ ( x \in X ) $$ is a continuous mapping of $Y$ onto $X$.    Now in Example 4.21 Rudin illustrates that compactness of $X$ is essential in Theorem 4.17.   So here is Example 4.21:   Let $X$ be the half open interval $[0, 2 \pi )$ on the real line, and let $\mathbf{f}$ be the mapping of $X$ onto the circle $Y$ consisting of all points whose distance from the origin is $1$, given by $$ \mathbf{f}(t) = ( \cos t, \sin t ) \ \ \ \ \ \ ( 0 \leq t &lt; 2 \pi). $$ Then this function $\mathbf{f}$ is of course continuous and injective. But Rudin asserts that the inverse function is discontinuous at the point $(1, 0) = \mathbf{f}(0)$. How to demonstrate this fact rigorously?  So far in the book, Rudin has not discussed the functions $\cos$ and $\sin$. So can we come up with an example that uses only what Rudin has discussed? Over the time I've been able to vote for closure on SO, I've gotten into the habit of clicking on the reason itself in order to select the radio button corresponding to it. Since the reasons have changed, there are also now links in the middle of the text, which I also happen to accidentally keep clicking on. More annoyingly, these links don't open in a new tab so I have to click back and redo it. This happens most often with the "FAQ" link in the off-topic dialog, which is right there in the top left corner where I normally click.        Basically, those pieces of text now have two functions. One function of it is selecting the corresponding radio button and the other function is loading a new page. Obviously loading the new page trumps simple selection.  This has become even more painful now with the new off-topic messages, where a large portion of the message is a link:        A simple feature request: can the vote-to-close dialog itself not render the links? Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$ This is a homework question whereby I am supposed to evaluate:  $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$  Wolfram Alpha outputs the answer as   $$\frac{1}{2}(\pi \coth(\pi) - 1)$$  But I have no idea how to get there. Tried partial fractions (by splitting into imaginary components), tried comparing with the Basel problem (turns out there's little similarities), nothing worked. Most efficient way to append arrays in C#? I am pulling data out of an old-school ActiveX in the form of arrays of doubles. I don't initially know the final number of samples I will actually retrieve.  What is the most efficient way to concatenate these arrays together in C# as I pull them  out of the system? Can you partition an infinite set, into an infinite number of infinite sets? How much longer will my 10 year old frame last? A short story about finding an inhabitable planet and meeting aliens So the story goes, humans in a spaceship search for an inhabitable planet, being surprised they found none in the vicinity of Earth. They find one, and it seems uninhabited. At the same time are observed by the aliens, who go down on the planet to meet them. The aliens sit them down and tell them a story about war and conflict that ruled the galaxy, and how all the sentient races in the galaxy got together and got rid of the race that tried to conquer all by blowing up their planets.  In the end you realise man is this terrible conqueror the aliens are talking about.