Interpolation of bounded sequences by $\alpha $-dense curves

In 1905 Lebesgue showed that there is a sequence of continuous functions, put $f_{n}:[0,1]\longrightarrow [0,1]$, which interpolates any sequence in $[0,1]$, that is, given $(a_{n})_{n\geq 1}\subset [0,1]$ there is $t\in [0,1]$ such that $f_{n}(t)=a_{n}$ for each positive integer $n$. This result was improved (in the sense of Theorem ) in 1998 by Y. Benyamini. In this paper, we generalize the Benyamini's result in Theorem . The key tool for this goal are the so called $\alpha $-dense curves. We apply our results to approach the solution of a certain infinite-dimensional linear program with a countable number of constraints