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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 ?

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