# Is it acceptable to use “But” in a proof that doesn't use contradiction?

I have recently read a lot of proofs that like to say "But..." right before the punchline. I feel that the word "But..." should be used if what follows is contradictory in some way, as in proofs by contradiction. However, many proofs that don't use contradiction simply use "But..." to introduce the punchline (the example that inspired this post is at the end of this proof of the Alternating Series Test. Is this considered acceptable form? It's so common but sounds strange to me.

• Looks fine to me. If you don't like the word "but" for some reason, don't use it. – André Nicolas Apr 13 '16 at 1:29
• I would use 'but' to indicate that something is somewhat surprising, in a loose sense. In the proof you link to, they are essentially writing that "$b_{k+1}$ is nonnegative but less than or equal to $a_{m+1}$" which I think is pretty standard English. It helps to give your reader a slightly better feel for what the important parts of the proof are, as well as breaking up the monotony of "and... and... and... " – Jair Taylor Apr 13 '16 at 1:33
• That proof is not really the best example, though, since it's not written in full sentences like you will generally see in papers. – Jair Taylor Apr 13 '16 at 1:34
• It's one way to alert the reader to a consequence of something that may have been stated or shown quite a lot earlier. Suppose statement $A$ is proved,and it implies $B$, but $B$ is not mentioned because it is not needed until 4 paragraphs later. Rather than asserting $B$ then, leaving the reader wondering where it came from, you may say, "But $A\to B,$ so...". – DanielWainfleet Apr 13 '16 at 2:04

I think that "but" can also be used to exemplify or to provide a unintuitive result (consequence) or more evidence that was not immediately obvious. For example:

claim a
evidence a
some more evidence....

but.. since we know that from evidence a... then we have
theorem b!


Or rather:

claim a
evidence a
claim b
..but notice that... x = y because of evidence a and claim a as assumed so...
claim b is true!


I also think that such usage is acceptable and is not necessarily required to be an introduction to a contradiction.

A common use seems to be

1. We have situation X.

2. We also have additional feature Y, that causes the situation X to assume a much simpler or different-looking form Z.

3. Z is different enough from X to call attention to the contrast between them with words like "but", "however", "actually", or "in fact".

The ability to draw such contrasts is useful even when Z is a special case of X and not in contradiction to anything.

Generally, if the initial idea of the proof changes as the narrative develops, maybe to handle some complication or special cases, some words like "but" or "however" may appear when indicating that.

In English, the word "but" can function two different ways:

• As "not this, but that", it signifies an exclusive or.
• As "not only this, but also that", it signifies a conjunction.

The word "but" by itself is often an abbreviated version of one of those two. In the usage you ask about, the latter is meant.