Sample: Proof of induction without proof of induction.
We know it's true for n equal to 1. Now assume that it's true
for every natural number less than n. n is arbitrary, so we can
take n as large as we want. If n is sufficiently large, the case
of n+1 is trivially equivalent, so the only important n are n less
than n. We can take n = n (from above), so it's true for n+1
because it's just about n.