Placement of "for all n" in "show that A(n) holds"

Question

Suppose that we have a statement A(n) in a mathematical text, where n is a number.

Where in "show that A(n) holds" should the phrase "for all n" be inserted?

• For all n[,] show that A(n) holds.
• Show[,] for all n[,] that A(n) holds.
• Show that[,] for all n[,] the statement A(n) holds.
• Show that A(n) holds[,] for all n.

About the commas, I feel that they are necessary in the first and second sentences, yet the phrase "for all n" is, in my opinion, quite important for the understanding of the whole sentence (e.g. should A(n) only holds for one n or no n?). Moreover, is it safe to leave out the first comma in the third sentence?

Answer 1

Regarding the commas: I would agree that "for all n" is central to the meaning of the sentence, so it is not parenthetical and cannot be set apart completely. The only comma that makes sense is the one following the prepositional phrase (and even this is not strictly necessary since the phrase is short). Any other commas surrounding the phrase indicate that it is non-essential, when in reality the question's objective relies on it. Thus:

• For all n, show that A(n) holds.
• Show for all n that A(n) holds.
• Show that for all n(,) the statement A(n) holds. (A comma can be placed after the phrase here, but to me the sentence flows better without it. No comma should be used before it.)
• Show that A(n) holds for all n.

Overall, I think the last option is the most straightforward in conveying the task at hand. They all roughly indicate the same thing . . . but if there is a semantic argument to be made, one could construe the first and second as asking the reader (solver?) to show that A(n) holds individually for every possible n, implying a laborious case-by-case proof, while the third and fourth make it clear that the task is simply to show generally that A(n) always holds.

Answer 2

Okay, I've gotten rid of all extraneous commas from your four given options.

• Show that for all n, A(n) holds.

Among your four options, this one most accurately and least ambiguously conveys the intended meaning.

However, the phrase "for all n" it is still very much Mathlish: there's a plural-singular disagreement, and the variable n is simultaneously acting as a common noun (n = integer(s)) yet conflictingly representing a proper noun (n = 37). Better:

• Show that for every/each object n, A(n) holds. ✅

• For all n, show that A(n) holds. ❌

• Show for all n that A(n) holds. ❌

Technically—and this point is subtle—these two sentences fail to accurately convey the instruction's intended meaning. For example,

• for every integer n, show that A(n) holds ❌

is literally assigning infinitely many tasks, which naturally can never terminate or be completed; in contrast, the existence of infinitely many integers doesn't preclude

• show that for every integer n, A(n) holds ✅

from being fully and correctly responded to in a single page (potentially using a mathematical-induction argument or reasoning by the logic of dominoes).

• Show that A(n) holds for all n.

This sentence's hanging quantifier renders it ambiguous between your other three options.

Answer 3

For all n[,] show that A(n) holds.

"for all n" is a prepositional phrase acting as an adverbial modifier to the whole of the main clause, which is "show that A(n) holds"

Show[,] for all n[,] that A(n) holds.

"for all n" is a prepositional phrase acting as an adverbial modifier to the whole of the main clause, which is "show that A(n) holds"

Show that[,] for all n[,] the statement A(n) holds.

"for all n" is a prepositional phrase acting as an adverbial modifier to the whole of the main clause, which is "Show that the statement A(n) holds"

In all of the above "for all n" is a sentence adverbial, otherwise known as a free modifier.

Free modifiers are invariably offset from the main clause by a comma (if fronting) or commas if contained within the main clause.

Unfortunately, he was ill.

He , unfortunately, was ill.

He was , unfortunately, ill.

Show that A(n) holds[,] for all n.

No comma is required as "for all n" is the complement of "Show that A(n) holds"