$k$-Weight Independent Set
Input: A vertex weighted graph $G=(V,E,w)$ and an integer $k$.
Question: Is the a set $V'\subset V$ such that $V'$ is an independent set and $\sum_{v\in V'} w(v) \geq k$?
Showing that this problem is in NP is trivial, but I'm suck on showing that $X \Longleftrightarrow Y$. The forward direction is simple I think.
$\Longrightarrow$ If you have a $k$-Independent Set, then we know that each node has a weight that is a natural number ( $\ge 1$ ), so the graph has weighted independent set of at least $k$.
But how would you show the reverse direction $\Longleftarrow$?