supremum and infimum understanding

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I looked at definitions and questions already posed, but none helps in clarifying this specific challenge I am trying to understand.

Assume x $\in$ X = {1,2,3,4,5,6,8}

and I want

  • inf{ x | x $\geq$ 5.5 } = 6
  • sup{ x | x $\leq$ 5.5 } = 5
  • inf{ x | x $\geq$ 8 } = 8
  • sup{ x | x $\geq$ 8 } = 8

Is this correct? What I'm wary about is that some might argue that

  • inf{ x | x $\geq$ 5.5 } = 5