bolzano

The Autonomous Province of Bolzano has a high level of self-government and a special fiscal regime.

Stuttgart to Bolzano via the Brenner pass (1370m) and back.

But the graph of 2023(?) data [0] varies hugely by region, from a TFR of 1.65 in Bolzano in the far north, to <= 1.0 in Sardinia and the central regions.

> Stuttgart to Bolzano via the Brenner pass (1370m) and back.

(Which they've quite plausibly got the right to do - Israel may have signed the Fourth Geneva Convention, but the West Bank wasn't claimed by another sovereign state at the time of its occupation.) European countries aren't nearly as generous when they win wars - which is why Gdańsk is Polish, Bolzano is Italian, Kaliningrad is Russian, and Hungary is a whole lot smaller than it used to be.

If that were true, everybody and their little sister would understand the Bolzano–Weierstrass theorem by now.

You can actually stop and look at Ötzi in Bolzano/Bozen if you're ever in the area.

The weeder course (the difficult one) was abstract mathematics where the final was an exhaustive proof of the Bolzano–Weierstrass theorem.

Bolzano proved (his version of) the intermediate value theorem in 1817, but it has been used in calculations for _way_ longer.

However, go and try to explain Bolzano's Theorem or the midpoint rule using that framework.

In some extreme cases, they even start with infinitely small sequences, with some rather gnarly theorems like Bolzano–Weierstrass theorem about converging subsequences.

Can't recommend it—and Bolzano, or Bozen, depending on whether you lean Italian or German—enough!

[2] "The province defying Italy’s birth dearth: Thirty years of child-friendly policies" describes how Bolzano - South Tyrol's child-friendly policies and $ incentives have given it the highest birthrate of all of Italy since the 1990s.

I might ask, “What are the most important theorems in calculus?” The second theorem which comes to mind is the Bolzano-Weierstrass theorem, which involves infinite sequences and is the foundation for many other core results in calculus.

Dedekind cuts (which are necessary for epsilon-delta proofs) or alternatively Bolzano Weierstrass theorem (if you use sequential convergence definition) imply multiple sizes of infinity, so the logical arrow is in the other direction, roughly: "real analysis implies assigning different sizes to sets", not "assigning different sizes to sets implies real analysis".

If that were true, everybody and their little sister would understand the Bolzano–Weierstrass theorem by now.

You can actually stop and look at Ötzi in Bolzano/Bozen if you're ever in the area.

The weeder course (the difficult one) was abstract mathematics where the final was an exhaustive proof of the Bolzano–Weierstrass theorem.