Find the lower bound of a contingent claim in incomplete market

I'm trying to justify the lower bound for the price of a contingent claim (a European one) which is not marketable in an arbitrage free market. I would like to have your advice on my way to do it:

First, I start with the set $$\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}$$ where $$V_0(\theta)$$ is the initial value of a portfolio with strategy $$\theta$$. I would like to show that $$C_{-}:=\sup{\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}}$$ is a lower bound for the pricing of the contingent claim. Indeed, consider a price $$V_0(\theta), then the buyer can short sell $$C_{-}$$, buy $$V_0(\theta)$$ and invest $$(C_{-}-V_0(\theta))$$ in a risk free asset such that at the maturity date $$T$$ the buyer can face the short selling by using $$H$$ and he receives $$(C_{-}-V_0(\theta))(1+r)^T$$ which is strictly positive.

However I am wondering if my arbitrage strategy is correct since why one will buy $$C_{-}$$ if there exists $$V_0(\theta) ?

I think I have misunderstood something Thank you a lot