Let the function $f : \mathbb{R} \to \mathbb{R}$ be differentiable and let $f'$ be a uniformly continuous function. For each $n \in \mathbb{N}$, let the function $f_n : \mathbb{R} \to \mathbb{R}$ be defined by the formula: $$f_n(x) = \frac{f(x+\frac{1}{n}) - f(x)}{\frac{1}{n}}$$ Show that $f_n \rightrightarrows f'$. Give an example illustrating the necessity of the assumption of uniform continuity of the function $f'$. I see that the $f_n(x)$ formula is a formula for derivatives of $f$. If it's not tricky I think that I have no problem to show that pointwise. However, I don't know how to prove the uniform continuity using the definition with $\epsilon$.
Let $f$ be a real valued, continuous and differentiable function on the real numbers with $f'$ uniformly continuous. Show that $g_n(x)$ with $$g_n(x)=nf(x+1/n)-nf(x)$$ is uniformly convergent to the derivative of $f$.   I cannot figure out how to solve this problem. If anyone could help me for this problem I would be very grateful.
How can I add a bookmark in my nautilus folder explorer In Ubuntu 11 I just dragged the folder there.  This seems to be not possible on my new Samsung ultrabook with Ubuntu 12.04   I've tried Ctrl+D and using the top menu but nothing happens and no error messages, just nothing.  How can I add a bookmark? For a project into balloon simulation I'd like to know how the force on a balloon changes with altitude:  I know that the Buoyant force on a balloon is:  $F = (\rho_{air} - \rho_{helium})gV$  Using changing air density from the international standard atmosphere and changes in gravitational acceleration I can calculate the buoyant force and add this to the (negative) force of gravity mg.  However I know that it isn't this simple, the balloon changes volume with altitude as the air pressure outside the balloon changes.  I'd like to know how the balloon inflates - this would be a great help!  I think it may be something to do with ideal gas laws -  I can calculate air temperature and pressure from the international standard atmosphere but do I use pressure as the air pressure? - or air pressure minus balloon pressure? Boot Camp error "Windows cannot locate the disk and partition" I have a 2019 MacBook Pro 16 inch running on macOS Big Sur 11.1. I am trying to install windows using Bootcamp. I get the following error: Windows cannot locate the disk and partition specified in the unattended answer file's  setting. Make sure the setting references a valid partition and restart the installation. Can someone help me how to solve this? Thanks in advance Prove that at most $2k+1$ independent sets form the set of vertices of a graph, where $k$ is the maximum degree. Horizontal spacing issue in \listoftheorems using amsthm + thmtools Is it okay to use my US visa after a name change? I had my US visa about 7 month age, and I had to renew my passport for certain circumstances, so there was a mismatch in the name between both the visa and the new passport, you see they had to put my grand grandfather's name which was not on the old passport nor the visa, the name on old passport and visa is: Abdelrahman Samy Ali Mohammed and the new passport is: Abdelrahman Samy Ali Ibrahim Mohammed. My grand grandfather's name is Ibrahim, so the question is that gonna cause me any troubles concerning my entry to the US? is there any document should I consider taking it while i'm traveling?  Thanks,, Proving which QM interpretation is correct Let's assume that the existence of gravitons is theoretically proven or they are detected by LHC could one interpretaion be proven correct? I would like to solve the beginner's standard exercise which claims that a point of $\mathrm{Spec} \ R$ is closed iff it is a maximal ideal.  The reverse implication is easy. The direct one seems much subtler. If $P = V(I)$ then, since there exist a maximal ideal $M$ containing $P$, it would follow that $\{P, M\} \subseteq V(I) = P$, whence $P=M$. In order to show the existence of $M$ I applied Zorn's lemma to the set of ideals containing $P$.  The question is the following: is there any other proof of the above implication that does not (indirectly) use the axiom of choice? My concern is that I may be using a cannon to shoot a fly. Also, it would be the first time that I see closedness of points to require the axiom of choice. If the axiom is indeed needed in general, are there "nice" classes of rings for which we could get away without it? $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$ Consider a finite-dimensional vector space $V$ over the field $\mathbb{F}$ with a linear operator $T.$ Prove that we have $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2 \operatorname{rank}(T^2).$   First, let us assume that $\dim_{\mathbb{F}}(V) = n.$ Our immediate observations include that $T^3(V) \subseteq T^2(V) \subseteq T(V)$ so that $\operatorname{rank}(T^3) \leq \operatorname{rank}(T^2) \leq \operatorname{rank}(T)$ and $\operatorname{rank}(T^3) \geq \operatorname{rank}(T) + \operatorname{rank}(T^2) - n$ by Sylvester's Rank Inequality. Can we do some manipulations with what we have to get the desired inequality, or do we require something else? How to make \left, \right pairs of delimiter work over multiple lines? I have a very long equation that must be separated into two lines, and it has several pairs of nesting \left \right delimiters. Unfortunately, it seems that they need to be on the same line in order for them to work.  Also I'm using the align environment because I need the aligning functionality. Is there a way to have the size of delimiters automatically adjusted over multiple lines? Pass an extra argument to a callback function The attached animated gif shows some frames of a video with a mesh box placed at each tracking marker used to successfully track (solution error &lt;.4) an 800 frame chunk of video.   The point is to illustrate just how poorly the camera tracker calculated: when the box is at a marker placed at the top of a faraway building on the left, it is actually visually much larger than when it is at the tracking marker closest to where the real camera was. I went to great lengths to be sure that the camera's sensor size and focal length are correct, and manually placed and followed all markers to be sure they were accurately tracking.   There is not a lot of parallax motion over the 800 frames, but there should have been enough to at least know that the point closest to the camera moved more than the point furthest away. Sometimes it seems to work, sometimes it doesn't. Does anybody know how to improve this? How to enable exFAT in Ubuntu 14.04 Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? How to find and replace in linux on a large file? Oh, hi; So I've got this big json file and I'm trying to find and replace a few things but when I've used sed I get an error and I need a sed or awk string to find and replace it, without getting an error.  prajwal  Current: {&quot;telephone_numbers&quot;:[&quot;1-  Future: {&quot;telephone_numbers&quot;:[&quot;+1-  This is the command i tried and the error i got. sed -i 's/{&quot;telephone_numbers&quot;:[&quot;1-/{&quot;telephone_numbers&quot;:[&quot;+1-/g' output.json sed: -e expression #1, char 56: unterminated `s' command Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller? When you add two independent normal distributions the resulting distributions' variance is the sum of the variances i.e. it gets larger.  However, the Central Limit Theorem states that  when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed.   and also says that as you add more variables the variance gets smaller. Why does the variance increase in one case but decrease in the other? Is it because in the CLT case the random variables come from the same underlying distribution? Dropping Internet connection on Realtek Semiconductor Co., Ltd. RTL8821AE I have continuous problems of network dropping connections on wifi with ubuntu 16.04, I have an ASUS R510J, this is my network card:  ##### lspci #############################  03:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821AE 802.11ac PCIe Wireless Network Adapter [10ec:8821]         Subsystem: AzureWave RTL8821AE 802.11ac PCIe Wireless Network Adapter [1a3b:2161]         Kernel driver in use: rtl8821ae  04:00.1 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 12)         Subsystem: ASUSTeK Computer Inc. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [1043:200f]         Kernel driver in use: r8169   sudo service network-manager restart doesnt's help, as I never regain internet connection after it drops, and I have to restart the PC. On Ethernet cable I have no problem.  What do you think is the issue? drivers?  thank you Disabling OBSOLETE error in C# How to redirect users to custom lostpassword page? Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks