Ethan Akin's "proof" that all vector bundles are stably trivial, and hence the $K$-theory of any space must vanish:
Let $V$ be a vector bundle over the base space $B$. Let $T$ be a trivial bundle of the same rank as $V$. To show that $V$ is stably trivial, it suffices to prove that $$V\oplus V=V\oplus T$$.
Let $P$ be the principal bundle associated with $V$. Pull $P$ back over itself to get a bundle $Q$:
Then $Q$ (together with the map to $B$) is the principal bundle associated to $V\oplus V$. But the bundle $Q\rightarrow P$ clearly has a section, namely the diagonal map (viewing $Q$ as a subspace of $P\times P$). Thus $Q=P\times GL_n$, which (together with the same map to $B$) is the principal bundle associated to $V\oplus T$.
