Let $(M,d)$ be a metric space and $x\in M$, $x_n\to x$  So $|d(x_n,a)-d(a,x)|\leq d(x_n,x)$ is a concluse form the triangle inequallty?
Here there is my proof (quite short and easy) of a rather straightforward result. The text of this question comes from a previous question of mine, where I ended up working on a wrong inequality. Here things are fixed. Still, I would like to know:  if it is sound, because absolute value always creates me some problem, and if there is a shorter (neater) way to get the result (maybe without using contradiction).    Proposition: $|d(x,y)-d(y,z)| \leq d(x,z)$. Proof: We proceed by cases.  Case 1: $d(x,y)-d(y,z) \geq 0$. Assume by contradiction that $d(x,y)-d(y,z) &gt; d(x,z)$. Hence, $d(x,y)&gt; d(y,z) + d(x,z)$, contradicting the triangle inequality.  Case 2: $d(x,y)-d(y,z) &lt; 0$. Assume by contradiction that $ - d(x,y) + d(y,z) &gt; d(x,z)$. Hence, $d(x,y) &lt; d(y,z) - d(x,z)$. By applying the triangle inequality, we have that $d(y,x) \geq d(y,z) + d(x,z)$, which is by symmetry equivalent to  $d(x,y) \geq d(y,z) + d(x,z)$, obtaining the desired contradiction. $\square$     As always, any feedback is welcome. Thank you for your time.
How to move an element into another element? Prove that "injective function $f:X\to Y$ exists" and "surjective function $g:Y\to X$ exists" is logically equivalent. Prove that   injective function $f:X\to Y$ exists surjective function $g:Y\to X$ exists   is equivalent.$(x,y\neq \emptyset)$   My approach  First,prove left to right.  If injective function $f:X\to Y$ exists, a set A can be defined like this. $$A=\{f(x):x\in X\}$$ then it is sure that $A\subset Y$, and $|A|=|X|$  And, if we define function $g$ like $$g(y)=\begin{cases}x &amp; \text{if $y\in A$ and } f(x)=y,\\x_0&amp; \text{if }y\notin A.\end{cases}$$ ($x_0$ is a fixed element of $X$)  Then, $g$ is a surjective function, so surjective function $g:Y\to X$ exists.  Now,prove right to left.  If surjective function $g:Y\to X$ exists, $\forall x \in X$, we can select a element $y\in Y$ that $g(y)=x$, which is clear that we can pick a different $y\in Y$ for $\forall x \in X$  if we define function $f$ as $f(x)=\text{one element in Y that g(that element)=$x$}$, f is a injective function, so injective function $f:X\to Y$ exists.    But, I am not sure that my approach is right, especially on proving right to left. Please check whether my approach is right, ways to improve it,  and some other ways of proving this question. How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work? How to resize partitions? How can I create a coffee cream effect with Photoshop? Today I would like to create a coffee cup with the cream floral on it as a text by using Photoshop. 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Explaining to laypeople why bootstrapping works Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$ Here is the question:     $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$   ()  I think we need to simplify it writing it in summation sign as you can see here:  $$\frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}}$$  or in Wolfram Alpha input in comments.  I can compute it too! It's easy to write a script for this kind of question. I need a way to solve it. How would you solve it on a piece of paper? 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I would ask my supervisor but I don't want it to imply that I'm avoid taking initiatives when I could do it on my own. Are there instances where citing Wikipedia is allowed? For example, suppose you're writing an article about triangles, and you want to include a short proof of Pythagoras' theorem, but you can't quite remember how to derive it, so you pop over to:  and check the proof there. It works mathematically, makes sense and is as far as you can tell, correct, so you include it. Now the proof on Wikipedia doesn't have a citation because probably someone has been taught it and copied it over to Wikipedia, but it's still correct. The only thing you could cite for the proof would be Wikipedia, would a citation be ok or should it just be copied down without a citation?  Edit: Ok, there have been many responses. Most with a firm "no", but some giving the reason: "Just get the proof from a book" since it's so common, but suppose then it wasn't a common proof of something. 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