What’s on the Supremum & Infimum poster
A definition and 6 facts worth remembering.
The supremum of a set is its least upper bound; the infimum is its greatest lower bound.
- sup S is an upper bound, and no smaller one exists
- sup S need not lie in S; a maximum must
- sup of the open interval (0, 1) is 1, not attained
- Every nonempty set bounded above has a sup in ℝ
- ℚ fails that: the sup of {x : x² < 2} is not rational
- By convention sup ∅ = −∞ and inf ∅ = +∞
- sup of (0, 1) — 1, not in the set
- max of (0, 1) — Does not exist
- inf of {1/n} — 0, never attained
- The sup of {x ∈ ℚ : x² < 2} is √2, which is exactly why ℚ is incomplete.
The least upper bound property is what distinguishes ℝ from ℚ. Every theorem about the real line rests on it somewhere.
Questions about the Supremum & Infimum poster
- What’s on the Supremum & Infimum poster?
- A definition and 6 facts worth remembering. The supremum of a set is its least upper bound; the infimum is its greatest lower bound. sup S is an upper bound, and no smaller one exists; sup S need not lie in S; a maximum must; sup of the open interval (0, 1) is 1, not attained; Every nonempty set bounded above has a sup in ℝ; ℚ fails that: the sup of {x : x² < 2} is not rational; By convention sup ∅ = −∞ and inf ∅ = +∞; sup of (0, 1) — 1, not in the set; max of (0, 1) — Does not exist; inf of {1/n} — 0, never attained; The sup of {x ∈ ℚ : x² < 2} is √2, which is exactly why ℚ is incomplete.. The least upper bound property is what distinguishes ℝ from ℚ. Every theorem about the real line rests on it somewhere.
- Who is the Supremum & Infimum poster for?
- Supremum & Infimum belongs to the Mathematics section rather than to a school year, because math is not something one grade owns. Anyone learning analysis can pin it up — a beginner, a student mid-course, or someone revising years later.
- When should you use the Supremum & Infimum poster?
- Any argument that needs a bound the set gets arbitrarily close to. A wall chart earns its place by being glanceable from where the work is happening, so Supremum & Infimum belongs on the wall where that math work actually happens, within glancing distance, rather than filed away.
- What other posters go with Supremum & Infimum?
- Absolute vs Conditional, Bolzano-Weierstrass and Cauchy Sequences sit alongside Supremum & Infimum in the Mathematics section. Printed together they make a wall rather than a single sheet, which is how a reference set actually gets used.Absolute vs ConditionalBolzano-WeierstrassCauchy Sequences
- Is the Supremum & Infimum poster free to download and print?
- Yes. Supremum & Infimum downloads as a free PDF with no account, no email and no watermark, like everything else in the Mathematics section. Print as many copies as you like for a home, a classroom, a library or a tutoring group; reselling the file is the only thing the licence rules out.Read the licence
- What size does the Supremum & Infimum poster print at?
- Supremum & Infimum is a vector PDF laid out for US Letter, and prints on A4 with Fit to page — the same file, no separate download. Because every mark on it is drawn rather than photographed, it stays sharp enlarged to A3, A2 or A1 at a copy shop. Colour carries emphasis only, so a greyscale print of Supremum & Infimum loses nothing.Printing guide
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