There's some evidence that precisely the _opposite_ can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One memorable instance is his _Exemplum Memorabile Inductionis Fallacis_, where he described how he was almost led to conjecture a recursive formula for a particular numerical sequence until he found that they disagreed on the 10th term. (There are other reasons for that formula to have been plausible; that and other topics are discussed in [this article](http://eulerarchive.maa.org/hedi/HEDI-2005-08.pdf "Ed Sandifer: How Euler Did It by Ed Sandifer. A memorable example of false induction").)

(Incidentally the "right" formula is [now quite well-known](https://www.jstor.org/stable/1990932 "George E. Andrews: Euler's Exemplum Memorabile Inductionis Fallacis and q-Trinomial Coefficients; Journal of the American Mathematical Society, Vol. 3, No. 3 (Jul., 1990), pp. 653-669").)