Supose that $f\colon [0,1] \rightarrow \mathbb R$ is a continuous function. Show that: $$\lim \limits_{n \rightarrow+\infty} \int_{0}^{1}{f(x^n)dx}=f(0)$$  I'm studying the Riemann Integral theory and I have not idea to how do this. I need some hint to start because I do not see how to apply the Riemann sum or other theory concept to be able to help me.
$$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$  when f is continuous on $[0,1]$  I know it can be proved using bounded convergence theorem but,  I wanna know proof using only basic properties of riemann integral and fundamental theorem of calculus and MVT for integrals ...   Thank you.
After my , I've now got a new suggestion: down votes should not have any side effect in meta.  By side effects, I mean something like banning me from asking questions due to too many down votes, which I am experiencing right now:     states that I am permanently banned unless I improve my standing by posting some quality answers.  Seriously, I'm new to the site, and thus I'm not capable of posting quality answers on Meta.  So my only refuge was to create a new account. I think this check for previously downvoted posts should be disabled on Meta. Is it bad to use a pacifier? I've been using it when she gets fussy and she seems to love it. Is this a bad habit to get into? I've heard there are potential dental/breastfeeding/dependency issues. Is there any truth to this? Do reroll and dice-adding abilities work on critical hits and critical failures? When a spaceship is experiencing a constant acceleration of $10m/s^2$, the astronauts will be moving at nearly the speed of light after about a year in the earth's reference frame. This means the spaceship's energy will start to diverge as a function of the speed $v$ so there will be a huge amount of energy necessary to increase the speed of the ship any further. This way, the speed of light can never be crossed.  All of this is clear to me, but all of this is also formulated in earth's reference frame. But from the astronaut's reference frame: the spaceship is simply accelerating at $10m/s^2$ and so the force on the spaceship is constant. Then why would we need huge amounts of energy to accelerate the spaceship?  For example, I read somewhere that the amount of energy that would be needed to accelerate a large spaceship to half the speed of light is more than 2000 times the current world annual energy consumption. How does this make sense in the astronaut's (non-inertial) frame? How does a wheelie work? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. How to find an end point of an arc given another end point, radius, and arc direction? How do people understand astromech droids? Include some chat capabilities in iOS app Currently, the app notifies users when they're mentioned in chat. However, when we tap on the notification, we got redirected out of the app to Safari.   Is there a chance we can get some chat capabilities in the app, so that we don't have to login again in Safari to quickly reply to the messages?    App version: 0.1.19 09/19/2014 Where does Minecraft for Windows 10 store its data? I have searched in %appdata%, pulling up the list of files there, but there is no file labelled .minecraft. Searching %appdata%\.minecraft returns "no results found".   I also opened Minecraft to click on the resources button and could not locate that either. Every time I click to open a link, other windows start opening. They all want to sell me Detox your Mac or have warnings about how my Mac is at risk or infected.  How can I check for malware? I upgraded my Nexus 4 to Lollipop a few days ago but I don't like it. I'll leave out my reasons because they're not relevant here. I wiped my phone and installed a clean stock 4.2.2 version instead. I'm happy with this.  Not long after, Android automatically downloaded the 4.3 upgrade and notifies me with the choices "Later" and "Install". I don't want to do this upgrade.  How do I:   get rid of the notification, and  remove the downloaded update (to free up precious storage space).     I have already tried choosing "Later" but the same notification immediately reappears.    Based on Google search results, I also tried going to Settings &gt; Apps &gt; All &gt; Google Services Framework and unticked "show notifications" but it still appears. Force stop or disable also doesn't seem to help, so I've re-enabled the service including notifications. I didn't see any other apps in this list that sounded relevant - did I just not see it? How many of Trelawney's prophecies have come true? Why do we say that irreducible representation of Poincare group represents the one-particle state? Only because   Rep is unitary, so saves positive-definite norm (for possibility density), Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and It is the representation of the global group of relativistic symmetry,    yes? Make WCF service be accessible through Internet I would like to know if the same Minecraft account used on a computer can be used on a different platform like the Xbox 360? I am not asking about playing on each simultaneously as each platform has been purchased - I am asking if you can cross play on multiple platforms with the same account. Basically I would like to know if all the saved stuff and settings from one platform will carry over to another. What do the "+n" values mean at the end of a method name in a stack trace? How do I identify which features have NULL attributes in the attribute table of a shapefile?  When i try [field]= 'NULL' it selects the features with value = 0 (see picture) Adding tags to files in OS X: How to use, where are tags stored, and will they work on a network? I want to check if points layer that I have intersect with polygons layer that I have, as Boolean column in the points dataframe. I have two GeoPandas dataframes, the first one is many points and looks like this: &gt;&gt;&gt;ID  geometry 0  12  POINT (5.0279 7.4547) 1  45  POINT (6.6539 12.139) ...  and the second dataframe is one layer of many different polygons  that looks something like this: &gt;&gt;&gt;name     code   geometry 0  Desert   12     POLYGON ((5.52013 13.8902, 5.5265 13.892,...) 1  Water    24     POLYGON ((5.53756 13.88472, 5.5291 13.8791,...) ...  I'm trying to check if there is any intersection between the points layer and the regions layer. For that I have determined the crs and then used intersects : regions=regions.to_crs({'init': 'epsg:4326'}) points=points.set_crs({'init': 'epsg:4326'}) inter=points.geometry.intersects(regions.geometry)  A The script runs with the following warnings:  FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6 return _prepare_from_string(&quot; &quot;.join(pjargs)) /opt/conda/lib/python3.8/site-packages/geopandas/base.py:39: UserWarning: The indices of the two GeoSeries are different. warn(&quot;The indices of the two GeoSeries are different.&quot;)  Then when I check the results the only value is False, like all th epoints do not intersect: inter.unique().tolist() &gt;&gt;&gt;[False]  *I saw on QGIS that the there are points that intersect and there are points that do not so there is no way this result is true *I have checked the dtypes - each one of my geodataframes has one column that is geometry and called geometry. My end goal: to add new column in the points geodataframe that will tell if it intersects the regions or not. I had this problem a week or two ago and I posted here but nothing from the suggested solutions seemed to work. I have managed to get some more details on it, in case anyone can help me, because this issue is causing me huge productivity problems and I really want it solved.  So I'm using Linux mint distribution, and I've installed Lyx via the software manager. Every time that I open a new document, I get an error claiming that some files are missing, and until they are installed, compiling the files or exporting them as pdfs is not possible. The file in question is called "extarticle.cls". Additionally, under Document->Settings->Document Class, every document class is marked as unavailable.  Here is the thing though.. this file is not missing... I have located it inside a folder associated with Lyx. It's definitely not missing.  I really need help with this one guys... It's been weeks since the problem came up and I really just want to be rid of it.. Why does this thing not work? Any help?