soft question - What are some slogans that express mathematical tricks?

"It is easy to prove existence when there is only one, or when there are many"

explanation:

If there is only one object with a certain property, you can sometimes use it to define it. For example, in geometric situations, you can sometimes define it locally and glue the patches since uniqueness guarantees compatibility on overlaps. It suggests that you should try proving uniqueness before proving existence and if uniqueness fails, maybe you should add constraints (thus, paradoxically, adding constrains can help in proving existence). On the other hand, sometimes it is easier to prove that there are many than to point out one specific example (transcendental numbers, continues nowhere differentiable functions,...). Therefore, you may want to seek for the right notion of "many" in your universe (cardinality, measure, "topological bigness" like the baire property,...) and try to prove that actually there are "few" objects that don't have the required property.

comment: This relates to the answer saying that when you can't avoid making a choice, make all of them simultaneously. This happens when there are more than one, but not many...