Let $h_1, h_2,$ etc. be a sequence of positive real numbers such that $$\sum_nh_n = \infty.$$ Let $x_1, x_2,$ etc. be a sequence of real numbers in $(0, 1)$.
Let $f_0, f_1,$ etc. be of sequence of functions $[0,1]\to [0,\infty)$. Define $f_0 \equiv 0$ and, for each $n$, let $f_n$ be the smallest concave function such that $$f_n(x_n) \ge f_{n-1}(x_n) + h_n$$ and $f_n \ge f_{n-1}$ pointwise. Does the Lipschitz constant of $f_n$, $$\sup_{x, y}\frac{|f(x)-f(y)|}{|x-y|},$$ necessarily tend to infinity?
Edit: Prior to this edit, the question was incorrectly formulated as "Does $\sup_x f_n(x)$ necessarily tend to infinity?"
