In the subset sum existance problem, we have a sequence of integers. We are given an integer $x$ that we should look for a possible subsequence that sum to $x$. For example, if we have the sequence $\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $x = 39$ then we have a subsequence $\{ 10, 9, 8, 7, 5 \}$ that sums up to $39$.
I want to prove that any sequence $[1, n]$ have a subsequence sum for any $x \in [1, n(n + 1) / 2]$. In the previous example, $n = 10$ so that any $x$ between $1$ and $55$ inclusive can be found. I have made a program and tested that this is true. Now I want to determine a proof.