Voldemort already knew he planned on making 7 Horcruxes - he asked Slughorn specifically about 7 and also the rocks on his window. So, why didn't he know Harry was one? Or surely he knew Harry would end up becoming one, inevitably meaning that he would die in the battle and life was pointless?
Harry: "He made seven Horcruxes?"   Dumbledore: "...Not seven Horcruxes: six. The seventh part of his soul, however maimed, resides inside his regenerated body..." (HP6, US Hardback, page 503).    I would think that Dumbledore had already known that there were 6 horcruxes:    the diary  Gaunt's ring  Salazar's locket Helga Hufflepuff's cup Nagini (the whole essence divided thing)   Now those are the five that he was sure of. He knew that when Riddle came to Hogwarts to ask for the DADA position, he was really there so he could hide a horcrux. (But he never got a chance to tell Harry that, right?) Did Dumbledore know of the diadem? Or did he just think Riddle wanted to hide the cup?   Did Dumbledore say there were 6 and not 7 because he didn't know about the diadem but did know about Harry being a horcrux? Or did he say there were 6 to ensure Harry didn't guess he was a horcrux? Sorry if this makes no sense.  I know that in the end there were 7 horcruxes and an 8th part of Voldemort's soul right?
Prove by induction that $$(a_{1}+a_{2}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)\geq n^{2}$$  where $n$ is a positive integer and $a_1, a_2,\dots, a_n$ are real positive numbers   Hence, show that  $$\csc^{2}\theta +\sec^{2}\theta +\cot^{2}\theta \geq 9\cos^{2}\theta$$  Please help me.  Thank you! I am going to Santa Monica for 4 weeks to polish my English in July and would like to volunteer at the Planetary Society located in Pasadena.  I am a Swiss citizen and have a biometric passport. However, I don't have any special visa, as there is no visa needed for the school I am attending and using the Visa Waiver Program, I am allowed to stay up to 90 days in the US.  Am I allowed to do volunteer work during my stay? Is it possible to view Google Chrome bookmarks and history from the terminal Is it possible to view Google Chrome bookmarks and history by using command line? Delete specific line number(s) from a text file using sed? Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$  Prove $(X,d)$ is a metric space.    My attempt:  This question comes down to showing that $d$ is a metric, which means that:   $d(f,g)\geq 0$ $d(f,g)=0\iff f=g$ $d(f,g)=d(g,f)$ Triangular sadness.     1) Nonnegativity is guaranteed by taking the supremum of the absolute value of a sum, since the absolute value of a sum is non-negative, the supremum of non-negative numbers is non-negative.  $\square$    2) For the supremum of $|f(x)-g(x)|$ to be equal to $0$, the greatest number for $|f(x)-g(x)|$ from $0\leq x\leq 1$ must be $0$, hence $|f(x)-g(x)|=0$ for all values on this range, and thus $f(x)=g(x)$ is the constant zero map.  $\square$    3) $|f(x)-g(x)|=|g(x)-f(x)|$ implies that $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|=\sup \limits_{0\leq x \leq 1} |g(x)-f(x)|$ $\square$    4) Triangular sadness:  We need to prove that $d(f,g)\leq d(f,h) + d(g,h)$  I.e.  $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|\leq \sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$  Now I am not sure how I can 'behave' with the sup 'function'. Can I do the following?  $$\sup|f(x)-g(x)| = \sup|f(x)-g(x)+h(x)-h(x)|=\sup|f(x)-h(x)+(h(x)-g(x))|$$ Then use normal triangular inequality:  $$\sup|f(x)-h(x)+(h(x)-g(x))|\leq \sup(|f(x)-h(x)|+|h(x)-g(x)|)=\sup(|f(x)-h(x)|+|g(x)-h(x)|)$$   This is obviously less than $\sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$, but I don't know how to prove it. How do I finish that one off. Any way to send a personal message to another user? Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look like...   For the first one, $xy=yx$ allows us to say that any element in the group can be written as $x^iy^j$ for some $i,j$. More precisely, using $x^5=y^3$, we can always reduce $j$ to $0,1$ or $2$, whereas $i$ could have any value. So I was thinking about $\mathbb{Z} \times C_3$. But the other way around (reducing $i$ this time) it's also $C_5 \times \mathbb{Z}$. But then I should be able to prove that both descriptions are equivalent, which I can't do...  For the second, I proceed similarly and get to the same kind of problem...  Could you tell me where I'm going wrong?  Thank you very much in advance! I am trying to prove the following statement but have trouble  comprehending/going forward with some parts! Here is the statement:     If $A$ is any set, then  $|A|$ $&lt;$ $|P(A)|$   Here is what I have so far:  We need to show that there is an injection from $A$ to $P(A)$ but not a surjection.   A natural choice for an injection is the function $ f(x)$ $=$ $\{x \}$, which in plain English, takes any element $x$ (that is in $A$) and sends it to the one-element set $\{x \}$. Thus $f(x)$ is injective!  To show that there is no surjection, for the sake of contradiction, assume there is a surjection. Here is where I start to have trouble. Surjectivity means that every element of the co-domain is mapped to an element of the domain, correct? Consequently, in this particular case, we are "matching" sets (from $P(A)$) to elements (from $A$) right?  If the above is correct, my problem arises here. I am not sure how to prove that $f$ is not surjective. Unfortunately, I am easily confused by notation so please explain in English.  Thank you in advance!! :) Can someone please confirm the following? I have a C1/D visa. I worked as a flight attendant (currently unemployed) and entered the US numerous times, and also visited Mexico in December 2017.  I am planning on visiting Mexico again for 3 weeks in due course. Will the C1/D allow for me to transit in the US and visit Mexico or do I need a B1/B2 visa? I had issues with CBP before as it seems to be a very grey area.  According to the Mexican Consulate in South Africa, the type of visa is irrelevant. As long as it is multiple-entry, I am good to go.  Can someone please confirm, as I cannot get a solid answer from anyone?  Thank you! From Euler we've learned that $z=re^{i\theta}$.  And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$.  Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different exponent $z=r\pi^{i\theta}$ and get the same size for $z$ as I did before: $|z|^2=r\pi^{i\theta}\times r\pi^{-i\theta}=r^2$  What did I miss? How can I upgrade Ubuntu LTS to the next one offline? I'm using Ubuntu 12.04 on my computer and I want to upgrade to Ubuntu 14.04.  I do not have internet at home, so I need to do the upgrade offline.  On openSUSE there a way to make the upgrade offline simple and effective way is by using  the iso DVD.  My question is: Is there any way to make Ubuntu upgrade offline as in openSUSE ? Why is my StackExchange OpenID so hard to find? Arriving at a different port of entry in the same state in USA? I am a girl in my early twenties traveling to USA from India and in my visa application I have provided the contact address for a relative in San Francisco, and their invitation letter. I would like to visit LA first. Will I face problems during immigration if I land in LA (with an immediate on wards ticket to SF)? I dont plan to get on the flight from LA to SFO.  Initially I was planning to fly to SFO from India and then fly to LA immediately after completing immigration at SFO airport. How to find unused/dead code in java projects How to theme the maintenance mode page? What would be the best practice for theming the maintenance mode page in Drupal 7? Really stuck with this, \begin{cases}  2tx'(t)-1=x^2t \\  x(t=1)=1 \\  \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :( How to get my own IP address and save it to a variable in a shell script? why does OSX add extra '._' when I tar a directory? I'd just like to know why if I "tar -czf" a file/directory, osx adds a:  ._  for each file? I see these when I untar in linux. Or when I work with the uncompressed project in eclipse as it doesn't like them at all. I use 10.7.5. Shortest path from points in one layer to points in another layer via roads in QGIS Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.