In Armstrong's topology book, he asks the reader to
Show that in a graph there is a subgraph which is a tree and which contains all the vertices of the original.
My Proof
Fact: Every graph must contain finite number of cycles.
Fact: Removing an edge always decreases number of cycles.
Till the graph has a cycle, remove any arbitrary edge from the cycle. This process preserves connected-ness, since the edge was part of a cycle. Since, at every step, number of cycles decreases, this process must terminate. The final result is a connected graph without cycles which contains all vertices, which is precisely a spanning tree. QED.
Is it correct?
