Luke Posted February 5, 2019 Share Posted February 5, 2019 (edited) On 2/5/2019 at 12:50 AM, DavidW said: If 1/infinite is shorthand for lim_(n->infinity) 1/n, then it is zero. Why? As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero? (I'm assuming you too have a background similar to mine and/or a degree/PhD in Mathematics, which is very likely...) Edited March 31, 2019 by Luke Quote Link to comment

grodrigues Posted February 5, 2019 Share Posted February 5, 2019 @Luke: ""As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero? " Your question is baffling because you give the answer in it. A limit of a sequence or function (that it is *the* limit, when it exists, is a theorem) is *by definition* a *real number* (let us stick to real analysis here) that satisfies a certain property, "the value" in your sentence. Period. *The* limit lim_{n->0}(1/n) *is* a real number, namely 0; adding "exactly" does no work and conveys absolutely nothing. \math Quote Link to comment

Luke Posted February 5, 2019 Author Share Posted February 5, 2019 @grodrigues Are you talking about the fact that properties should not be used to give definitions? Quote Link to comment

DavidW Posted February 5, 2019 Share Posted February 5, 2019 2 hours ago, Luke said: Why? As far as I know (Mathematical Analysis I course, engineering curriculum), a limit is the value that a function or sequence approaches as the index approaches some value. So, how can it be exactly zero? (I'm assuming you too have a background similar to mine and/or a degree/PhD in Mathematics, which is very likely...) The definition of the limit of a sequence x_0, x_1, x_2, ...x_n, .... is that number x (if there is one) such that for any e>0, you can find some n so that all the sequence members after x_n are within e of x. so if our sequence is 1, 1/2, 1/3, 1/4, ..., 1/n, ... then zero satisfies the definition (the sequence gets arbitrarily close to it) so it’s the limit. It’s fairly easy to show that if a series has a limit, it’s unique. Note: we’re not saying that the sequence eventually gets to 0: any member of the sequence is non-zero. But zero is still the limit, i.e. the value that sequence elements get arbitrarily close to. Quote Link to comment

grodrigues Posted February 5, 2019 Share Posted February 5, 2019 @Luke: "Are you talking about the fact that properties should not be used to give definitions? " No. where did you get that idea? What I was saying, or tried to say, was (1) by *definition* the *limit of a sequence*, when it exists, is a *real number* (once again, sticking to real analysis) satisfying a certain property that David gave (in symbols, the sequence x_n converges to x if (by definition) A e > 0 E n_0 \in N A n > n_0: |x_n - x| < e with N the set of natural numbers, A the universal quantifier and E the existential one) and (2) in your very sentence you speak about the limit as " the value that", so unless you are using the words in a total novel way, it is baffling (at least to me) why you are asking "how can it be exactly zero?" \math Quote Link to comment

Luke Posted February 5, 2019 Author Share Posted February 5, 2019 18 minutes ago, grodrigues said: @Luke: "Are you talking about the fact that properties should not be used to give definitions? " No. where did you get that idea? From the fact that I didn't state the definition of a limit but just a property..... Sorry if I misunderstood your words...... Anyway, it seems both you and @DavidW know a lot about math, I may ask you clarifications about some theorems (maybe ....) Quote Link to comment

Luke Posted March 31, 2019 Author Share Posted March 31, 2019 (edited) @DavidW, @grodrigues Do you perhaps know how to make this "box" in LaTex when the document class is article? EDIT: OK, don't worry, I found them...... I was looking for something like this: \begin{tcolorbox}[colback=blue!5!white,colframe=blue!75!black,title=My title] My box with my title. \end{tcolorbox} Edited March 31, 2019 by Luke Quote Link to comment

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