To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term):
One way to prove that a mathematical object $O_1$ has some property $P$ is to:
- Construct a (topological) space $\mathfrak M$ containing the object $O_1$ such that $P$ is a continuous invariant in $\mathfrak M$, meaning that if $t \mapsto O_t$ is a continuous path of elements of $\mathfrak M$, every element $O_t$ has property $P$ as soon as one of them has it.
- Prove that some element $O_0$ in $\mathfrak M$ has property $P$.
- Prove that $O_0$ and $O_1$ are linked by a path $t \mapsto O_t$ in $\mathfrak M$ (for example, prove that $\mathfrak M$ is path-connected).
This strategy is for example well illustrated by a very nice proof of the genus-degree formula.
My question is do you know any example of an application of this strategy in more elementary mathematics?
After all, even in high school or in the first years of college, the mathematical world is full of elements living in connected spaces (points, lines, triangles, numbers, functions...) and I see no reason that this proof strategy couldn't work in this context, even without the topology jargon.
Of course, one can imagine variants of this strategy (restraining oneself to polygonal paths, for instance) and I'm interested in all of these. However, I would like the answers to keep a topological flavour: I'm not interested (in this question) by examples using other kinds of invariance (e.g. through the action of a group).
A last remark: I haven't found this strategy in problem-solving books so I don't know if it has a well-established name. If you know of such a name, or of a book mentioning this strategy, please tell me!