How to prove that any number from $0$ to $\frac{n(n+1)}{2}$ can be obtained as sum of subset of set {$1,2,3, ... ,n$}

By induction. Easy to confirm for small $n$.

Suppose the claim is true for $n-1$, so we can get every number from $0$ to $\frac {n(n-1)}2$ just using $\{0,1,\cdots, n-1\}$. Now we add the element $n$ and we seek to get the numbers from $\frac {n(n-1)}2+1$ to $\frac {n(n+1)}2$. But $$\frac {n(n+1)}2-\frac {n(n-1)}2=n$$ so subtracting $n$ from any of those numbers gets you into the previously solved range, so we are done.