Suppose you are trying to prove result $X$ by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result $X'$ (that you don't really care about) by induction. This has a chance of succeeding because you have more to work with in the induction step. My favourite example of this is Thomassen's beautiful proof that every planar graph is 5-choosable. The proof is actually pretty straightforward once you know what you should be proving. Here is the strengthened form (which is a nice exercise to prove by induction),

**Theorem.**
Let $G$ be a planar graph with at least 3 vertices such that every face of $G$ is bounded by a triangle (except possibly the outer face). Let the outer face of $G$ be bounded by a cycle $C=v_1 \dots v_kv_1$. Suppose that $v_1$ has been coloured 1 and that $v_2$ has been coloured 2. Further suppose that for every other vertex of $C$ a list of at least 3 colours has been specified, and for every vertex of $G - C$, a list of at least 5 colours has been specified. Then, the colouring of $v_1$ and $v_2$ can be extended to a colouring of $G$ with the specified lists.

**Question 1.** What are some other nice examples of this phenomenon?

**Question 2.** Under what conditions is the strategy of strengthening the induction hypothesis likely to work?