let $k$ be an integer, and consider the set $S$ of functions from $\mathbb{N}$ to $\mathbb{N}$ where $g(x) =0$ for all $x&gt;k$. I want to show $S$ is countable.   This is what I have so far, but I am not quite sure if I am going about it correctly, or what to do next.  So our functions will look like this   $A_1=\{(1,a_1),(2,a_2),(3,a_3),(4,a_4),....,(k,a_k), (k+1,0), (k+2,0)....\}$   It seems from here I should be able to show that there are only countable ways of doing this, but I am not quite sure how to say that. I think the best way would to come up with a bijection from the naturals to $S$ but for the life of me I can't come up with one.  I would love some help. Thanks in advance.
Prove that $\mathbb{N}^k$ is countable for every $k \in \mathbb{N}$.  I am told that we can go about this inductively.   Let $P(n)$ be the statement: “$\mathbb{N}^n$ is countable” $\forall n \in \mathbb{N}$.  Base Case: $\mathbb{N}^1 = \mathbb{N}$ is countable by definition, so $\checkmark$  Inductive Step: $\mathbb{N}^{k+1}$ $“=”$ $\mathbb{N}^k \times \mathbb{N}$  We know that $(A,B)$ countable $\implies$ $A \times B$ is countable. I am stuck on the part where I have to prove the rest, but I know that, for example, $(1,2,7) \in \mathbb{N}^3 \notin \mathbb{N}^2 \times \mathbb{N}$ but instead $((1,2),7) \in \mathbb{N}^2 \times \mathbb{N}$. So how would I go about proving the statement.
How to reference sections, equations, and figures in a separate document (e.g., supplemental information)? I have a main document (manuscript) and a supplemental document.   I wonder if there is a way that I can "link" the two (make them recognize each other) so that section, equation, and figure references across documents are recognized? The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:     Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not invertible, then $D/(x)$ is isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.   Now the exercise asks:     Prove that $A$ is not a Euclidean Domain.   Everything hints to an argument by contradiction: let $(A, d)$ be a ED and $x$ an element of minimal degree among the non invertibles we'd like to show that $A/(x)$ is not isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.  How do we do that? My problem is that, since I don't know what this degree function looks like, I don't know how to choose this $x$!  I know that the elements of $A/(x)$ are of the form $a+(x)$, with $a$ of degree less than $x$ or zero. By minimality of $x$ this means that $a\in \{0, 1, -1\}$. Now I'm lost: how do we derive a contradiction from this? Isomorphism between dual space and bilinear forms Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. How can I copy a custom Dictionary on Word 2000? How to represent it with UNIQUE indexes and foreign keys? "Group" is a group of pupils. "Topic" is a topic of lessons.  To every group corresponds exactly one topic. (So there may be multiple groups per topic.)  To a pupil corresponds multiple groups, but maximum one group per topic.  How to implement this in terms of UNIQUE indexes and foreign keys in MySQL? Finite Index of Subgroup of Subgroup Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.  So this is basically a bijective proof? The number of cosets of $H$ in $G$ equals the number of cosets of $K$ in $G$ times the number of cosets of $H$ in $K$ by the multiplication principle?   Reference: Fraleigh p. 103  Question 10.35 in A First Course in Abstract Algebra Lyircs of Free by Natalia Kills:     I'm free      I just spent all my money       but I rocked that like it don't cost a thing   Shouldn't it be "I've just spent all my money"? As far as I remember, our English told us to use Present Prefect in such case because of the just. Sorting list based on values from another list Determining likely relationship based upon autosomal DNA shared cM Lots of questions in the reopen queue Does removing an application from Software Center do a remove or a purge When I remove an application already installed through the Software Center. Does it remove the app the same way as doing a purge to it (aptitude purge) or does it remove it like a "remove" (aptitude remove). Does any test actually prove causation? Installing drivers for windows? Archimedes' principle for two liquid layers Problem: I have a cylindrical vessel of height $H$ and radius $R$. There are two liquid layers in the vessel. The first has density $D_1$ and height $h_1$, the second has density $D_2$ and height $h_2$. The second liquid is floating on the first liquid (thus $D_2 &lt; D_1$) and they are both entirely within the vessel ($h_1+h_2 \le H$). Now I have a cube of density $D$ and edge length $n$. The cube fits into the vessel ($n \le \sqrt2R$) but it might not be completely inside it ($n&gt;H$ is possible). I place it into the vessel such that one side is horizontal (the cube is not lopsided). Some liquid might overflow as a result. How do I use Archimedes' principle to calculate the topmost liquid level after placing the cube? I can ignore damping and other physical effects. EDIT: Note that the only values I know are $H, R, D_1, h_1, D_2, h_2, n$ and their abovementioned constraints.  My workings: I tried using the formula directly, where $\text{weight of cube}=\text{weight of displaced liquid}$. I first get $Dn^3=D_2n^2s_2$ where $s_2$ is the height submerged in the second liquid. If $s_2 \le h_2$ then the cube is only submerged in the second liquid. So the answer is $\min(H, h_1+h_2+{n^2s_2\over\pi R^2})$, accounting for overflows. This case is simple. But if $s_2&gt;h_2$ then the cube can be submerged in the first liquid too. I thought of subtracting the weight of the second liquid displaced to obtain the height submerged in first liquid, but realize that I don't know that. The weight displaced depends on the final liquid height, which can be affected by overflows, whether the cube fits totally within the two liquids, etc. In fact, it can be the case that after the liquid level rises, the cube now displaces more of the second liquid, causing it to not be submerged in the first (is that even possible?) It seems very messy and I have no idea how to start. :( Does anyone have a nice solution to this problem? I need to draw a tree, and constantly using dotty and including the PNGs is starting to be a pain. Is there a way to get LaTeX to draw it's own trees that doesn't involve learning an entirely new language like TikZ? Let $X$ be simply connected and let $ \tau $ be a cycle in $X$. If $ \tau(0)=\tau(1)=x_0$ and $\gamma $ is a path between $x\in X$ and $x_0$, is $\tau$ homologous to $ \gamma\tau\gamma^{-1} $?  I think it is, but I'm not sure about the proof. Can I use the fact that $ \gamma\tau\gamma^{-1} $ is homologous to $ \gamma+\tau+\gamma^{-1} $, where $+$ is the operation between 1-chains (that is abelian)? Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you. Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$   I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help? This is a phrase I've heard many people use, and it sounds wrong to me; e.g.:     The thing about that is, is that she might take it the wrong way.   It seems to treat "The thing [...] is"—the entire phrase—as the subject of the sentence. Of course, even if that were allowed, the comma placement would still be questionable.  I'm curious to know:   Is there a name for this wording? Is it specific to a certain dialect? Is its etymology known? Are the rationals a closed or open set in $\mathbb{R}$?