# Proofs

A question that comes up a lot in class is, “how do you get good at proofs?” (It’s usually framed as “I don’t like proofs”, but we’re not having any of that negativity *here*, thank you very much.)

I don’t have a silver bullet for that. I do have some advice.

### Keep a notebook

In your special proofs notebook, keep track of all the tricks that seem obvious in retrospect - or that make you say “I’d never have thought of that!” When you’re stuck, flick back through it and see if anything jumps out as a possible way forward.

### Do lots of proofs

I can’t explain the process of walking - when I was a kid, I just sort of kept falling over until I got passably good at it. Similar thing with proofs: try. Fall over. Try again.

The more you do, the easier it gets.

### Get a feel for things

Try to get a feel for the kinds of problems that tend to use one method or another. After a while, you develop a sense of smell for it.

### Look for nuggets

I find that many proofs have a “nugget” - a key flash of insight in the middle of routine work. Focus on those nuggets. Respond to them with an ‘oo!’.

### Use the details

If a question gives you details - for example, that the function isn’t defined when $x=a$, or that your integer is odd, or similar, ask questions like:

- “Why have they told me that?”
- “What does that definition
*mean*?” (often if helps to write out definitions explicitly)

### Convince yourself, then prove it properly

Proofs are full of all sorts of rules about keeping things equivalent and what you can or can’t do. But those are for *presenting* proofs. When you’re trying to work out the way to a proof, it’s absolutely fine to disregard those rules and focus on convincing yourself.

You can tighten up the presentation later - but playing around is usually how you get to the nugget.

Do you have any top tips for generating proofs? I’d love to hear them in the comments.