Putnam 1961 B5

Let $k$ be a positive integer, and $n$ a positive integer greater than $2$ . Define $$f_{1}(n)=n,\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.