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 Jun10 asked Is Courant's Introduction to Calculus and Analysis still up-to-date? Jun9 awarded Nice Question May31 accepted Why to see that $\overline{B}(x;r)$ is closed if it was just defined? May30 awarded Popular Question May28 comment On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? What about the choice for $a$ in $|a_n-a|<\epsilon$? In some sequences, I can find an $a$ intuitively, I'm not sure if there is an algorithm to find it. I've tried to do the following: $$|a_n-a|<\epsilon\\-\epsilon < a_n-a< \epsilon\\-\epsilon -a_n<-a<-a_n +\epsilon\\\epsilon +a_n>a>a_n -\epsilon$$ But I don't know how to proceed from here - presuming that this is a way to do it. May28 accepted On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? May27 awarded Popular Question May27 asked Why to see that $\overline{B}(x;r)$ is closed if it was just defined? May26 revised On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? edited title May26 comment On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? So basically I must look for some $\epsilon$ that $\forall\epsilon\gt 0,\exists N\in\Bbb N :\forall n\ge N:|a_n-1|\lt\epsilon$ does not hold? The question seems to be utterly trivial, but I was thinking that some algorithmic process should be applied in order to automatically obtain those $\epsilon$'s. May26 revised On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? edited body May26 asked Are there functions that converge to $P$ when $f:\mathbb{N}\to\mathbb{R}$ and to $Q$ when $f:\mathbb{R}\to\mathbb{R}$ with $Q\neq P$? May26 asked On the thought process for choosing $\epsilon$'s to check the convergence of $(-1)^n$? May19 accepted $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. May14 awarded Notable Question May12 comment $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. @T.Bongers $h$ is just an aribtrary name for the vector. When I used it, I was trying to refer to $v$, to avoid confusion with the $v_n$'s. May12 comment $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. @T.Bongers I guess I got it. $h$ is the vector $(c_1,c_2,c_3)$. Then I multiply each of it's coordinates for one vector of the basis in order to obtain $v$ which is: $c_1v_1+c_2v_2+c_3v_3$? May12 comment $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. @T.Bongers Any orthonormal basis is a basis for $\mathbb{R}$? I'm still a little confused because I've read a little about subspaces and there are subspaces that are different of the primary vector space. May12 comment $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. @T.Bongers I don't understand where are you taking the $c_n$'s from. I'm thinking like this: $v=(v_1,v_2,v_3)$, $v_1=(v_{1,1},v_{1,2},v_{1,3})$, $v_2=(v_{2,1},v_{2,2},v_{2,3})$, $v_3=(v_{3,1},v_{3,2},v_{3,3})$. May12 comment $\mathcal{B}=\{v_1,v_2,v_3\}$ is an orthonormal basis, find all the vectors $v\in \mathbb{R}^3$ such that $\langle v, v_1+v_2-v_3\rangle=0$. @T.Bongers I'm still stuck. I've written $\langle v,v_1 \rangle+\langle v,v_2 \rangle + \langle v,v_3 \rangle=0$ but I still can't see it.