# Show that $S_0$ is the smallest value of a super hedging strategy for a Call option in a arbitrage free market

I attempt a proof :

We want to show $$S_0=\inf\{V_0 : \exists\theta\;s.t\;V_T(\theta)\geq H\}$$

Suppose this is not the case. There exists $$V_0^{*}$$ such that $$V_0^{*}< S_0$$ and $$V_T^{*}(\theta)\geq H$$. Then, one can sell the contingent claim at price $$S_0$$ and buy $$V_0^{*}$$ in order to invest $$S_0-V_0^{*}>0$$ in the risk free asset, then at time T he has $$V_T^{*}(\theta)-H + (S_0-V_0^{*})(1+r)>0$$. Thus, we have created an arbitrage opportunity which is contradictory since we are supposed to be in an arbitrage free market.

Is this seems correct to you ?