Consider the Bernoulli Inequality, for instance. Basically it says the following:

For all $x\in\mathbb{R}$ such that $x\geq -1$ and all $n\in\mathbb{Z}^+$, we have that $(1+x)^n \geq 1+nx$.

I'm concerned about the equivalent ways to rephrase this result, and if they are correctly written. I'm gonna put here two of them.

1) Let $x\in\mathbb{R}$ be such that $x\geq -1$ and let $n\in\mathbb{Z}^+$. Then $(1+x)^n \geq 1+nx$.

2) If $x\in\mathbb{R}$ is such that $x\geq -1$ and $n\in\mathbb{Z}^+$, then $(1+x)^n \geq 1+nx$.

In the item 1), including a period and writing Then after it is grammatically correct? I see a lot of professors and books writing like this, but this Then is a conclusion from the hypothesis given. I always feels this is wrong in some way, but I'm never sure. The item 2) follows the format if P, then Q. In this case it's very clear, we don't use a period. It's precisely because of this that I feel strange about the first item, but there I didn't use any if, so maybe this is ok. I just want to be sure about the proper way to write this things.

Thank you.

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    $\begingroup$ The first one is grammatically okay. Starting a sentence with Then is alright in this case because the previous sentence is indeed a complete thought in its own right. Had the first sentence begun with If, the period would indeed be grammatically incorrect. $\endgroup$ – ervx Apr 15 '16 at 20:57
  • $\begingroup$ @ervx Thank you very much. That is what I wanted to hear. $\endgroup$ – Integral Apr 15 '16 at 21:03
  • $\begingroup$ Some people say that the first expression is incorrect, See for example "The grammar according to West", math.illinois.edu/~dwest/grammar.html#letthen. However one can often enough see it in renowned papers, and I suppose it has become customary. $\endgroup$ – Arthur Sinulis Apr 15 '16 at 21:06
  • $\begingroup$ @ArthurSinulis Now you got me confused... $\endgroup$ – Integral Apr 16 '16 at 13:08
  • $\begingroup$ I like to avoid using so many symbols, writing "let/suppose $n$ is a positive integer" instead of $n \in \Bbb Z^+$ looks better, in my opinion (this is just a comment on style, so feel free to ignore it if you disagree). $\endgroup$ – YoTengoUnLCD Apr 16 '16 at 21:13

I'm actually a strong proponent of versions like (1) - at least, some of the time.

In a complicated theorem, both the hypothesis and conclusion might be a fair mouthful. As a reader, I'm happy to have them more cleanly separated by a period. Rules like "don't begin a sentence with 'then'" are (to my mind) only valuable insofar as they lead to clear writing, or prevent confusion - I care much more about whether I can follow the statement of a theorem than I do about whether it matches with Strunk & White.

For an example of such a complicated theorem: consider something of the form

If $x$ is such that, for all $x<y<2x$, if $y$ is prime then $gcd(x, y+1)>1$, then $x$ is odd.

(Ignore the fact that this "theorem" is bonkers.) The fact that the hypothesis itself contains a conditional makes this really awkward to express as a single sentence; I'd prefer it broken in two. And there are definitely theorems this complicated, or more complicated, so while this concern might not arise frequently, it will arise a noticeable amount of the time.

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