Existence of a Non-negative Integrable Random Variable with Supremum-Constrained Survival Function

Let $( X_1,\dots,X_n,\dots )$ be a sequence of non-negative integrable random variables, Denote by $S_i(t) = \mathbb{P}(X_i \geq t)$, the survival function of $X_i,  i \ge 1$


Does there exist a non-negative and integrable random variable X with survival function S, such that $ \sup_nS_n(t) \leq S(t), \, \forall t \ge 0$?


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Erdos Erdos
The answer is accepted.
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