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I'm new to this site

I'm starting to learn how to write proofs correctly and I will be very grateful if you help me find and correct the places where I write the syntax in the wrong way.

Here's a basic theorem of limits, how correct is the writing, are there any needs to write more explanation of how I came to some conclusions?

$$\forall\alpha,x\in \mathbb{R}\left((\lim \alpha = 0,\lim x = \pm\infty) \Rightarrow \lim \frac{1}{x} = 0,\lim \frac {1}{\alpha} = \pm \infty, \lim \frac {\alpha}{x} = 0 \right)$$ $\large\forall \varepsilon,M\in \mathbb{R^+}\exists\alpha,x\in \mathbb{R}\Big((\lim \alpha = 0,\lim x = \pm\infty) \Rightarrow |\alpha| < \varepsilon, |x| > M \Big) \\ \large\Rightarrow |\frac {1}{\alpha}|> \frac {1}{\varepsilon}, |\frac {1}{x}| < \frac {1}{M} \\ \large\Rightarrow \forall \varepsilon,M,N,\epsilon\in \mathbb{R^+}\exists\alpha,x\in \mathbb{R} \left((|\frac {1}{\alpha}|>\varepsilon > N, |\frac {1}{x}| < M < \epsilon) \Rightarrow \lim {\frac {1}{\alpha}} = \pm\infty, \lim {\frac {1}{x}} = 0\right)\ (1.a) \\ \large\Rightarrow \left(\lim {\alpha} = 0, \lim {\frac{1}{x}} = 0 \right) \Rightarrow \left( |\alpha| < \sqrt{\varepsilon}, |\frac {1}{x}| < \sqrt{\varepsilon} \right) \Rightarrow \left( |\frac {\alpha}{x}| < \varepsilon \right) \\ \large\Rightarrow \lim {\frac {\alpha}{x}} = 0 \ (2.a) \\ \large \Rightarrow (1.a),(2.a) \hspace{1cm} \blacksquare$

First of all I'm not quite sure in what places should I write the "for all" and "there exists" at the start of each line, and is this a correct way to write marks (like (1.a)) at some lines so I can link them later as implications that support the explanation.

In general how strict is the writing of proofs, like should I include "for all" in all the lines?

  • There are a few problems here. The biggest problem is that the the statement you're trying to prove isn't clearly stated. What does $\lim \alpha = 0$ and $\lim x = \pm \infty$ mean ? Do you mean $\lim_{\alpha \to 0} \alpha = 0$, or somthing else entirely ? Once you've fixed this problem we can move on to to other issues. – Digitallis Aug 08 '20 at 21:15
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    The best general advice I can offer is: use way more words in your proofs! Think about your favorite textbook, the one that's easiest to read and gain understanding from. I bet that textbook uses far more words than symbols in its writing. You should aim to do the same. Math symbols are precise and concise, true, but there's more to communication (indeed, even understanding one's own writing) than brevity. – Greg Martin Aug 08 '20 at 21:28
  • Thanks for both of you answers, I'll try to use more words. Regarding the proof, alpha is an infinitesimal quantity and x is infinite –  Aug 08 '20 at 22:27
  • I think I understand now. In the book I'm reading it is just "lim alpha = 0" maybe it's because it's already known that alpha -> 0... The same goes for x –  Aug 08 '20 at 22:38

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