*Proof*.
Assume proposition is false and let

be a minimal counterexample. Let

be a non-trivial closed path in

, for instance the longest closed path.
Now

must be even, so

is even. Hence each connected
component of

is Eulerian as

is minimal. But then

is Eulerian: you can walk along

and include all edges of connected
components of

when encountered -- giving a contradiction.
Hence there are no minimal counterexamples.