I am interested in nested radicals of the form $$f(n,k) = \sqrt{n - \sqrt{(n+1) - \sqrt{\cdots - \sqrt{k-1 - \sqrt{k}}}}}$$where $n,k$ are positive integers with $n < k, k \geq 5$. It appears that if $n < m$,then $f(n,k) < f(m,k)$
I have tried messing around with the recurrence $f(n+1,k) = n - f(n,k)^2$, but couldn't find any way to use this to prove my observation.
Does anyone have any suggestions?