Applying Definition of Supremum

If a point in a set is an upper bound, it's the supremum

This week a student asked about where to start on a problem related to proving the statement shown above. The hard part was knowing how to apply the definition of what it means for α to be a supremum. Two properties need to hold. First, α must be an upper bound (which is given). Second, no other upper bound is smaller than α. Here an example of another upper bound β is shown. By explicitly noting these two parts, we are able to prove the statement as follows.

Although a shorter proof can be crafted, it may be no surprise that terse writing (e.g. as in Baby Rudin) leaves newcomers with much head scratching. At a high-level, the approach taken above is to 1) state the definition of what must be shown and then 2) verify each of the parts in the definition individually. This approach may essentially be applied recursively since, to show the second part of the supremum definition, we defined what it means to be an upper bound and then used that formula to continue.

Here’s a YouTube video walking through this solution.

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