How does following set have a supremum?
$$S = \bigcap_{n=1}^\infty \left[-\frac1n,1+\frac1n\right]$$
My doubt is that, since we are approaching $1$ from the right side there shouldn't be any supremum.
2026-03-28 05:45:46.1774676746
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Lowest Upper Bound or Supremum
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Every subset of $\mathbb R$ that has an upper bound also has a supremum. Your set $S$ is bounded above, for example, $10$ is an upper bound for $S$. Therefore, $S$ has a supremum. You are therefore wrong.
To actually figure out what the supremum should be, first think about what it cannot be.
Can the supremum be $0.5$? Why not? Can it be $1.5$? Why not?
The fact that $1$ is not a supremum (or even an upper bound) for any of the intervals you intersect doesn't stop it from being the supremum of the intersection as a whole.
The set $S$ is $[0,1]$. Plain and simple. The exact details of how we construct $S$ do not change which set we end up with. The nature of the boundary of $S$ at $1$ isn't "different" as a result of the intersections, and the same can be said about the boundary at $0$.
The set $[0,1]$ has supremum $1$ and infimum $0$.