| Coming up with examples || || Come up with some examples of objects that fit the definition. Emphasize edge cases. || Examples help to train your intuition of what the object "looks like". || For monotone increasing functions, an edge case would be the constant function.
| Coming up with counterexamples || || As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. ||
| Writing out a wrong version of the definition || || || || See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).