In one question I found that when water was heated from 0°C to 15°C then the volume 1st decreases and then increases why is that so ?
How can the Anomalous Expansion of Water from 4$^\circ$C to 0$^\circ$C be explained with reference to subatomic particles?
What is the difference between == and === in php. I am unsure when to use both.  Updated note: So that it shows up in StackOverflow search, the difference between == and === is the same as the difference between != and !==. Problem  The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:   $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$.   Evaluate the limit $\lim_{n\to\infty} a_n$.  Solution   Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n &lt; a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 &lt; \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n &lt; a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} &lt; \sqrt{2+a_{n+1}},$$ that is $a_{n+1} &lt; a_{n+2}$ by definition. Find bounds for $a_n$. The statement $$W(n): 0 &lt; a_n &lt; 2$$ holds for $n=1$, that is $0 &lt; \sqrt2 &lt; 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 &lt; a_n &lt; 2.$$ Adding two and taking square roots, we have that $$0 &lt; \sqrt2 &lt; \sqrt{2+a_n} &lt; \sqrt4 = 2.$$ The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$. Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.) We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$. Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 &gt; 0$.   Is my solution correct? Dwellers return automatically after finding a low amount of items in Wasteland variable access outside of if statement Repeat String - Javascript How to Tail -f output with colors using only Awk and show the rest of the output? I was able to do a tail -f output with colors using the awk command. But I'm unsure how to output the rest of the log as it seems to be filtering only those 2 lines. I also want to see the other lines as the log is written.  What do I need to put in the awk command to show the rest of the output?  Image of my issue. The red text above is without using awk and displays the entire tail -f log. And below that is the same command with the awk color filtering. Why do we need to set swap space as twice big as our physical memory? When we want to setup a linux system, there is a common a suggestion to set the swap space  twice as big as your physical memory. I want to know why do we need this, and how it is this suggestion come to be? Clipping a GEE Image with Python API I am looking for a way to clip image collection by a given polygon geometry with Python API, The GEE JavaScript equivalent is simply ImageCollection.clip(geom) () With the python API I get the error: 'ImageCollection' object has no attribute 'clip'.  I'm adding my code after using .mean() as a reducer My code: set the boundary polygon my_boundary = ee.Geometry.Polygon([ [[-5, 40],[65, 40],[65, 60],[-5, 60],[-5, 60]]] ,proj=None) #set the ImageCollection and Image img_Collection = ee.ImageCollection('COPERNICUS/S5P/NRTI/L3_NO2').filterDate('2020-03-01', '2020-03-27').select('NO2_column_number_density') my_Image=img_Collection.mean().clip(my_boundary) #Create Ipyleaflet map my_map = ipyleaflet.Map(zoom=7) my_map.add_layer(layer=my_Image) my_map error: AttributeError: 'Image' object has no attribute 'model_id' How to watch live youtube stream using vlc? Is it possible for now to watch live youtube video streams using vlc? I just don't want to use flash player. I know that vlc can play youtube videos using link for the page where that video is located, but how about live youtube videos? GetFeatureInfo from Image WMS Adding vertical space at the start of a page I'd like to, first thing before writing anything on a page, add some vertical space.  Minimal (non)working example:  \documentclass{article} \begin{document}  \vspace{250px} I want to be further down on the page :(  \vspace{250px} Haha! Sucker :D \end{document}     How can I get the first \vspace to work, just like the second one? While playing Phase Ten, is it legal to pick up the top card of the discard pile and then immediately discard that exact card? How do I get list of all tables in a database using TSQL? If every element of set A ia an element of set B then we say that A is subset of B. A is said to be proper subset if A does not equal B.  My question is - whether the symbol we use to show subset is  same as that for proper subset? I have seen that A is subset of A itself is shown by the same symbol as that for the subset while when we write improper subset, we usually write equal to sign with symbol of subset. So, please help me to know this. What is the behavior of "delete" with stack objects? Generating data with a given sample covariance matrix Make Blender usable on high resolution screen Effect of switching response and explanatory variable in simple linear regression How can I get an email message's text content using Python? This is almost surely an Arzela-Ascoli question, since it comes from an old exam of which such problems are quite common. Unfortunately, I can't seem to get it though.  $\{ f_n \}$ is a sequence of functions $[0,1] \to \mathbb{R}$ satisfying $|f_n'(x)| \leq \frac{1 + |\ln (x)|}{\sqrt{x}}$ and $-10 \leq \int_0^1 f_n(x) dx \leq 10$ for every $n$. The question is to show the existence of a uniformly convergent subsequence.  In these kinds of situations, the bounds given usually turn into something nice  to force uniform boundedness. Maybe I could hope to establish equicontinuity using the integral somehow, but I'm not told that the $f_n$ are always positive, so the absolute values would probably screw things up anyway.  Is it an easy problem I'm missing or is this totally the wrong track? If so, what's the right track?