USAMO 1991 Problem 3

Define the function $\mathrm{f}$ on the natural numbers by $\mathrm{f}(1)=2, \mathrm{f}(\mathrm{n})=2^{\mathrm{f}(\mathrm{n}-1)}$. Show that $\mathrm{f}(\mathrm{n})$ has the same residue mod $m$ for all sufficiently large $n$.