Exercise :
We consider a financial market of one period $(\Omega,\mathcal{F},\mathbb{P},S^0,S^1)$, where the sample space $\Omega$ is finite and the $\sigma$-algebra $\mathcal{F} = 2^\Omega$. Furthermore, $S^0$ is the risk-free asset with initial value $S_0^0 = 1$ at time $t=0$ and interest rate $r>-1$ (which means that $S_1^0 = 1+r)$ and $S^1$ is the risky asset with initial value $S_0^1 >0$ at time $t=0$ and value $S_1^1$ at time $t=1$ which is a random variable. Furthermore, we consider a buying right with payout $C=(S_1^1 - K)^+$ with exercise value $K>0$ and maturity time $T=1$. Let $\overline{\pi}(C)$ be the no-arbitrage upper bound for $C$. Show that : $$\overline{\pi}(C) \leq S_0^1$$ and that the equality holds $$\overline{\pi}(C) = S_0^1$$ if it is also assumed that $\text{ess} \sup S_1^1 = \infty$ and $\text{ess}\inf S_1^1 = 0$.
Attempt :
We know, that :
$$\overline{\pi}(C) = \sup_{\mathbb{Q} \in \mathcal{P}}\mathbb{E}_\mathbb{Q}\bigg[\frac{C}{1+r}\bigg]=\sup_{\mathbb{Q} \in \mathcal{P}}\mathbb{E}_\mathbb{Q}\bigg[\frac{(S_1^1 - K)^+}{1+r}\bigg]$$
But, note that :
$$\mathbb{E}_\mathbb{Q}\bigg[\frac{(S_1^1 - K)^+}{1+r}\bigg] \leq \mathbb{E}_\mathbb{Q}\bigg[\frac{S_1^1}{1+r}\bigg] = \frac{\mathbb{E}_\mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
Thus, indeed it is :
$$\overline{\pi}(C) \leq S_0^1$$
Question : I would like to request some help proving the equality. I am really not very familiar with essential infimum and supremum and I would really appreciate a thorough elaboration or explanation over the specific example.