Why do people use "it is easy to prove"?

Use of "it is easy to see that" is common and traditional in mathematical writing, but it is not exactly a proud tradition: really good mathematical exposition uses this and similar locutions very sparingly.

To be more specific, I think it is bad writing to say "It is easy to see that X is true" and say no more about how to prove X. If this occurs in formal mathematical writing and all else is as it should be, then no information is being conveyed. In other words, what other reason is there to assert that X is true and say no more about the proof except that the author expects the reader to be able to supply the details unassisted? If you are skipping the proof for any other reason, you had better say something!

[And of course some of the harm is psychological. If you're carefully reading a text or paper that asserts X is true and says nothing else about it, you know you need to stop and think about how to prove X. If "it is easy to see X" then after every minute or so part of your brain will quit thinking about how to prove X and think, "They said it was easy to see this, and I can't see it at all. What am I, stupid?" I definitely remember thinking this way when I started out reading "serious" math books.]

When I referee papers I often suggest that authors suppress their "it is easy to see that"'s. As others have said in the comments, as a careful, skeptical reader, you also need to stop and be sure that indeed you can supply the proof yourself, and it is notorious among mathematicians that such phrases are likely places to find gaps in mathematical arguments. But it is just as easy -- in fact, easier -- to have a gap in the argument where you don't have any text at all, so writing "it is easy to see that" is not really the guilty party but rather a possible piece of incriminating evidence.

So if it's not so good to write this, why do people write this way? And they certainly do: I happened to be editing my commutative algebra notes when I read this question, so out of curiosity I searched for "easy" and found about ten instances of "it is easy to see that" in 265 pages of notes. About half of them I simply took out. The other half I thought were okay because I didn't just say "it is easy to see that": I went on to explain why it was easy! So having caught myself doing what I said not to, I can reflect on some causes:

1) Tradition/habit.

I have read "it is easy to see that" thousands of times, so it is in my vocabulary whether I like it or not. Most mathematicians know that they have funny phrases which appear ubiquitously in mathematical writing but not in the rest of their lives: one of the very first questions I answered on this site was about the meaning and use of "in the sequel". In the year or so since then I have observed it in my own writing: it just fits in there. You have to really actively dislike some of these standard locutions in order to avoid writing them yourself. For instance, I have more than a thousand pages of mathematical writings available online and I challenge anyone to find "by inspection" anywhere in these. "By inspection" is the deformed cousin of "it is easy to see that": whereas at least it is easy to see what "it is easy to see that" means, even the meaning of "by inspection" is obscure.

2) A conflation of formal writing and informal writing / speaking / teaching.

The way you speak mathematics to someone else is very different from the way you write it: it is much more temporal. If you are teaching someone new mathematics then most often they cannot verify / process / understand every single mathematical statement you make, in real time, so they have to make choices about exactly what to think about as you're talking to them. In spoken conversation it's extremely useful to say "this is easy": by saying it, you're cueing the listener that it's safe to direct her attention elsewhere. Also, because when you talk -- or write informally -- you don't give anywhere near as complete information as you do in formal mathematical writing, commentary on what you're skipping becomes more important. For instance, in an intermediate level graduate course I may prove approximately 2/3 of the theorems I state in class. If I'm skipping something, it's probably because it's too easy or too hard. I had better say which it is!

3) Immaturity/Laziness.

Certainly when you're reading your own writing and you find "it is easy to see that", you need to stop short and make sure you know exactly what you omitted. If it's not easy for you to see what you wrote it was easy to see, you may have a serious problem: indeed, you may be papering over a gap in your argument. To do this intentionally is a sign of great mathematical immaturity -- someone who hides (in plain sight!) what they don't know in this way is not going to make it very far in this profession -- but even doing it unintentionally is something that most mathematicians largely grow out of with experience.

Writing a paper is usually a long and tedious process. Some arguments seem (to the author) to be straightforward, but potentially painful to write out (e.g. due to having to introduce additional notation or concepts that are not relevant to the larger thread of the discussion at hand). At this point the author may simply state the required result, saying that it is "easy to prove", or something similar.

Ideally, such a result will indeed be easy to prove; e.g. at the level of difficulty of an exercise in a graduate text-book. Note that if your mastery of the subject is such that you are challenged by exercises in graduate-level text-books, then you may well find "easy to prove" statements hard to prove, not easy! The intended audience for such a statement is typically another expert in the field, not a beginner.

On the other hand, one reason that an argument can be hard, or at least tedious, to write down is that the author may not have at hand good tools for formalizing their intuition about the argument. In this case, rather than going to the trouble of developing these tools to formalize their intuition, they may just state the result, writing "it's easy to prove" or something similar. In my experience, this is usually why mistakes can creep in at these points --- because the difficulty in formalizing the intuition may be caused by an actual failing in the intuition!

The lesson I take from this in my own writing is that, when one thinks that a certain proof will be easy but tedious, one should examine the situation carefully, to make sure that the difficulty in writing out the complete argument is not being caused by some hidden flaw in one's intuition about the situtation.

As a reader of mathematics, especially if you are not an expert and are reading well-known papers that have been certified as correct by experts, it's probably best to presume that everything is in fact correct. However, one should expect that reading a paper and filling in all the details will be at least as demanding as reading a chapter in a graduate text-book on the same topic and doing all the exercises.

I like Pete's answer a lot, but I feel I should add something that I don't think he mentions, which is an unapologetic word of sympathy for the phrase. Here's the thing: I don't think "easy" means "easy to construct", it means "easy to follow". Coming up with the proof might not be trivial, but the process of coming up with a proof and the proof itself are (as anyone in mathematics knows) nothing like the same thing. Which of these is important to you will of course depend hugely on your level of mathematics and your reason for reading the text.

Having gone through undergraduate mathematics relatively recently, I've found myself picking up phrases like "and trivially we see that..." from my lecturers. It's a nasty habit, but on the other hand, I know what I mean when I say it, and by extension I know what they mean when they say it. I certainly don't mean to imply that my undergraduate career has been one long triviality, or that anyone who can't follow my train of thought is an idiot. I mean that, if I wrote the proof down in front of you, and maybe you did a bit of scratching about, you would find it followed trivially from previous work or a definition. I don't need to assume you are comfortable with this previous work or definition yet, and perhaps I even expect you not to be, unless I'm trying to teach in "real time". Some things take a while to sink in, and maths is not a spectator sport.

It's actually pedagogically very reasonable, in some cases. In fact, I would argue that not omitting some proofs is pedagogically very bad practice. Here are a few reasons, all of which apply hugely to undergraduate texts, but do apply elsewhere too.

1. Sometimes adding huge and reasonably elementary proofs distracts students from the main goal of the discussion.
2. Sometimes the proofs aren't very enlightening. Few people read a proof thinking "I must check this to see whether it's right". We read a proof to see why it's right, in order to learn from it, for our own selfish gain. If it's right because of something we already know, it is of little to no value to someone trying to learn from a text.
3. Anything I read in a textbook, especially a well known textbook, I (by gut instinct) assume to be important in my understanding of the content, otherwise it should have been consigned to an appendix (or the bin) for me not to read at my leisure. Eminent authors are authority figures in this way.

The example that sticks in my mind is a proof that the Ackermann function is not primitive recursive. The proof is a very simple concept and some very simple arithmetic manipulation, and your average 15-year-old could follow it without any trouble. But is it useful? Not in the slightest. It is one cute idea (namely "show that everything primitive recursive grows more slowly"), followed by several pages of very elementary mathematics which is so full of 'tricks' that it is very difficult to come up with, it is very difficult to memorise, and it takes half an hour to write down. And the tricks are so specific and elementary (things like "now replace n by the inequality n < 2n + 3") that they can't be used elsewhere. What's the value in that to someone learning recursion theory?