Worked solution: Euler's degree argument (1736)
The pairing argument just given proves one direction outright: an Eulerian trail can never have more than two odd-degree landmasses, since only its start and finish can end up with a leftover bridge. What takes more work — and what Carl Hierholzer nailed down rigorously in 1873 — is the reverse direction: that having or odd-degree vertices is not just necessary but actually enough to guarantee such a trail exists, with a construction that never leaves a bridge stranded.
The previous step already establishes the 'only if' half of Euler's criterion directly: if a connected graph admits a walk crossing every edge exactly once, then the number of vertices with odd degree is either (a closed circuit, if the walk returns to its start) or exactly (an open trail, whose two odd-degree vertices must be its start and its finish).
The converse — that or odd-degree vertices is also sufficient, for a connected graph — is the harder half. Euler asserted it in his 1736 paper without a complete proof; Carl Hierholzer supplied a full, constructive argument in 1873, showing how to build such a walk by stitching together closed loops (Biggs, Lloyd & Wilson 1976, Ch. 2).
Together the two directions give an exact test — — that applies to any connected graph whatsoever, which is exactly why this is a genuine theorem settling every possible bridge layout, and not merely an observation about one city.