What is the correct notation for increasing a limit in steps? [closed]

I want to increase my limit in steps, is lim(floor(n->infinity)) correct notationally, or should I just replace all usages of n with floor(n)

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It's not necessary to encode every detail of context in a mathematical expression. Surrounding text is your friend. For instance, you can write

$$\lim_{n\to\infty}\cdots$$ Here, the limit is taken over natural $n$.

If you're only ever working with integers or naturals in your discussion, then it'll probably be obvious that your limits do, too. To eliminate all doubt, you could write something like "Here, and throughout, limits are taken over natural $n$" with the first instance.

On the other hand, if you're presenting a mix of continuous and discrete limits in a paper, you can establish early on the convention that you'll use certain variables (say, $x$, $y$, $z$) for the former and others ($n$, $m$, $k$) for the latter (we sometimes also see $p$ and $q$ for primes). Also consider sprinkling-in confirming descriptors here and there, explicitly referring to "real $x$" and "natural $n$", etc, in your prose.

On the third hand, if you're mixing types of limits, and there's danger of expressions being read out of context and misunderstood, then it's perfectly acceptable to fold the domain into the symbol, as suggested in @TomKern's answer. You want to balance such notation against readability.

On the extreme hand, mathematics grants you the freedom to invent custom symbols to suit your needs. It's almost-certainly overkill here, but to distinguish continuous and discrete limits you could use, say, "$\lim$" vs "$\operatorname{dlim}$" (or whatever). Of course, with great power comes great responsibility: Care must be taken to clearly define such symbols and to not overwhelm the reader with too many of them.

On any hand, you can and should ask others for feedback on the comprehensibility of your work, from argumentation to presentation ... and that includes notation. Publications, in particular, often have guidelines about best practices (and/or editors to alert you to bad ones).