User:IssaRice/Boundedness theorem for continuous functions

There are two typical proofs of the boundedness theorem:

• one uses just the least upper bound property
• the other uses some sequences machinery (Bolzano-Weierstrass theorem, and continuous functions preserve sequence limits)

I previously thought these proofs were unrelated, or at least I didn't really think of how they might be connected.

Thinking about this more though, I think they have a similar kind of "mindset". Let's suppose we have a function defined on a closed interval, and suppose for a contradiction it is unbounded. We want to try to "climb the function" by taking bigger and bigger values. In the sequence proof, we pick $x_{n}$ so that $f(x_{n})\geq n$. But there could be multiple unbounded hills, so that's where the Bolzano-Weierstrass theorem comes in (it ensures that the $x_{n}$ values converge, which means we're basically climbing a single hill). If there was a way to pick out a single hill, we could just say "pick some x values in a way which climbs this hill, so that as we pick more values, we get closer and closer to the (non-existent) peak".

In the least upper bound proof, we also want to climb a hill. Not just any hill -- the leftmost hill. And the way we climb the hill is different. At every point, we ask ourselves "is everything to the left of me still bounded?" If yes, we nudge a bit rightward, and keep asking the same question. Once we get a "no" answer, we are in an interesting place (the same sort of place that the sequence proof is trying to find). There exists some "points shooting toward infinity" to my left, and yet my current place is bounded. If I nudge even a bit to the left, I get shot up to infinity, which means the function "breaks" at this spot.