Prove that  $ n &lt; 2^{n}  $ for all natural numbers $n$.  I tried this with induction: Inequality clearly holds when $n=1$. Supposing that when $n=k$, $k&lt;2^{k}$. Considering $k+1 &lt;2^{k}+1$, but where do I go from here?  Any other methods maybe?
I am currently in the process of learning how to prove statements by induction. For the most part, I understand but I am at most clueless when it comes to prove some inequalities.  For example, I am having a hard time proving that: $n &lt; 2^n$ is true. (Even if it's clear that it is indeed true!)  In the induction step, I am stuck with: $n + 1 &lt; 2^{n + 1}$  I don't know what would be the assumption I should make from this statement.  Anyone could point me in the right direction? Thanks!
Understanding the definition of the order of an entire function in Ahlfors's Complex Analysis Let $f: \mathbb C \to \mathbb C$ be an entire function. The order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r}, $$ where $$M(r)=\max_{|z|=r} |f(z)| .$$  Ahlfors in his Complex Analysis claims that     "According to this definition $\lambda$ is the smallest number such that       $$M(r)\leq e^{r^{\lambda+\varepsilon}} $$ for any given $\varepsilon &gt; 0$ as soon as $r$ is sufficiently large."   Why is this true?  My attempt:  We know that $$\lambda=\lim_{\rho \to \infty} \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}. $$  From the definition of the limit we have that for any $\varepsilon&gt;0$, there exists some $\rho_0&gt;0$, such that $$\left\lvert \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}-\lambda \right\rvert \leq \varepsilon ,$$ for every $\rho \geq \rho_0$. In other words $$\frac{\log \log M(r)}{\log r} \leq \lambda+\varepsilon $$ for every $r \geq \rho_0$. From here it is easy to see that $$M(r)\leq e^{r^{\lambda+\varepsilon}}, $$ for all $r \geq \rho_0$. I cannot see why $\lambda$ is the smallest number with this property.  Thanks in advance. What word or phrase best communicates the idea that a human (or other sentient being) is easily swayed, particularly between moral good and evil? Words I have found that are close are:   Impressionable - Easily influenced because of a lack of critical ability.   Implies that the subject lacks critical skills, and misses the concept of good and evil.  Gullible - Easily persuaded to believe something   Misses the concept of good and evil    I created a typical angels and demons scenario to demonstrate the polar extremes of the the spectrum. In this scenario, I lack the word to describe the humans, who can be influenced towards either end of the spectrum if impressed upon by an external entity.     The angels are devoted to good, and are immovable in their ideals. The demons are wicked, and will feign morality only to manipulate others. Both prey upon the humans, who are ____________.   I think that the key here is the spectrum of good an evil. Here are examples of other spectra and their associated words, for comparison.   Easily pushed from calm to angry: Temperamental Easily moved from determined to hopeless: Weak-willed Easily influenced by propaganda: In Bash, suppose I visit a directory, and then another directory. I would like to copy a file from the first directory to the second directory, but without specifying the long pathnames of them. Is it possible?   My temporary solution is to use /tmp as a temporary place to store a copy of the file. cp myfile /tmp when I am in the first directory, and then cp /tmp/myfile . when I am in the second directory. But I may check if the file will overwrite anything in /tmp.  Is there something similar to a clipboard for copying and pasting a file? Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$ Prove that $\sum_{k=0}^{m}\dfrac{\binom{m}{k}}{\binom{n}{k}}=\dfrac{n+1}{n+1-m}$.  We know, n > m. From the right side. we have $\dfrac{n+1}{n+1-m}=\dfrac{1}{1-\dfrac{m}{n+1}}$. since n > m. $0&lt;\dfrac{m}{n+1}&lt;1$. Then $\dfrac{1}{1-\dfrac{m}{n+1}}=1+\dfrac{m}{n+1}+(\dfrac{m}{n+1})^2+...$ I don't know what to do next? Did I choose the right path? Let $f \colon \mathbb{R} \rightarrow  \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how to use them here. Thanks. Segmentation fault on large array sizes It is common to see things described as the &quot;gold standard&quot;. It means that something is the best and is the ideal against which other things are to be compared; it cannot be superseded. I often , where a certain diagnostic test (e.g., the nasopharyngeal swab PCR test for infection with the Coronavirus causing COVID-19) is considered the most accurate and then other tests (e.g., cheaper or faster tests) are then evaluated against this benchmark. The etymology is covered in . In short, it is derived from the . Thus gold was the thing against which the value of all other currencies was measured. I don't like this term and would like to avoid using it. First, the desired meaning is not predominant; if you  you will mostly get results for the monetary system, not the more general phrase. Second, the etymology is outdated; the original meaning of the term now has the opposite connotation. The monetary gold standard was abandoned in the 1970s. Economists consider the gold standard to be a bad monetary system. Thus, if you are interested in economics and history, the term &quot;gold standard&quot; actually is associated with being outdated and flawed -- which is close to the opposite of the intended meaning. (Of course, many people are unaware of the original meaning of the term. ) (A note in response to comments: Why I want to avoid this term is relevant, but not essential, to the question. If you don't see a problem with the term, the question can be answered regardless.) My question: Is there a better phrase to use in place of &quot;gold standard&quot;? For example, what term could be used in a medical study comparing a new diagnostic test against the gold standard test? I'm not aware of any good substitute that has essentially the same meaning. Here are two example sentences where I would like a replacement:   A nasal swab is the gold standard for diagnosing COVID-19.     The  served as the gold standard for all scales from 1889 until 2019.   The best substitute I can think of is &quot;official,&quot; as in &quot;a nasal swab is the official way to diagnose COVID-19&quot;. &quot;Official&quot; implies some form of uniqueness. But &quot;official&quot; implies the existence of some authority that gives it this designation (as opposed to scientific consensus). A  asks for a &quot;perhaps humorous&quot; alternative -- the suggestion there, &quot;mother of all&quot;, is not appropriate here. (There is also .)  In response to the answers and comments doubting the premise of this question: I personally have a negative association with the phrase &quot;gold standard&quot; because I know people who advocate for returning to the gold standard or its modern equivalent, , and have discussed the topic at length. Several  US  continue to advocate for a return to the gold standard. Thus asserting that no one will interpret the phrase in terms of monetary policy is simply not convincing. What ultimates can be damage boosted by Mercy? What ultimates can be damage boosted by Mercy?  We already know that , but are there other ultimates that can be damage boosted by Mercy? Can all ultimates be damage boosted by Mercy? Cannot locate Ubuntu Software Center via Firefox to open APT links Convert a list to a data frame Profitable strategy in coin tossing? Why would parametric statistics ever be preferred over nonparametric? How to print colored text to the terminal? How can I output colored text to the terminal in Python? Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? @ variables in Ruby on Rails fsck error on boot: /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY I dual boot Ubuntu and Windows 7. I was browsing on FireFox in Ubuntu and all of a sudden I get some error message, so I rebooted.  Now when I try to boot into Ubuntu I get this message:  fsck from util-linux 2.26.2 /dev/sda6 contains a file system with errors, check forced. /dev/sda6: Inodes that were part of a corrupted orphan linked list found.  /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY.         (i.e., without -a or -p options) fsck exited with status code 4 The root filesystem on /dev/sda6 requires a manual fsck  Busybox v1.22.1 (Ubuntu 1:1.22.0-15ubuntu1) built in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs) _   How can I fix this? SQL Server CE in a multi user scenario - is this really, really stupid? Why on-shell vs. off-shell matters? The definitions between on- and off-shell are given in .   Why is it so important in QFT to distinguish these two notions? Gauss's law not making sense If we have a point charge and outside of it we have a non-conducting Gaussian sphere, then Gauss's law says that the net flux should be zero. I agree that the total field lines coming in are equal to the number of field lines coming out, but isn't it true that the magnitude of the electric field at those segments of the sphere where the field is coming in is greater than those where it is coming out? So when we integrate, wouldn't these areas, having a greater electric field, contribute more to the net flux? I don't see why they would cancel each other out. How to include audio file in PDF How do I include audio file, or should I say, link to audio file in PDF?   Audio should be playable "from PDF directly", without opening any additional window. Is it possible to attach audios to PDF     then so I can manipulate only with     that file, without being bothered     with copying folders where those     audio files would reside?   I saw that 1. is definitely doable (but don't know how) and I'm interested if second one is also possible (while compiling .tex file) since I know that some files (like graphs and tables) can be attached in PDF, but after it's generated.   Button for playing audio from document that uses this trick.