Transfinite induction is a process which extends ordinary induction beyond the natural numbers, for certain types of inductive hypothesis. It was first formulated by Cantor, inspired by a question about Fourier series put to him in 1869. This background is not very well known and quite surprising, as the relevance of transfinite induction to Fourier series is far from obvious. I will discuss this and some places in undergraduate courses where transfinite induction might be lurking. If time permits, I will include an instance where my own research took a route similar to Cantor’s although I did not know the historical background at the time.