Whether you consider it a deep, very deep or shallow result:
Theorem. The symplectic group ${\bf Sp}_{2n}({\mathbb R})$ is included in ${\bf SL}_{2n}({\mathbb R})$.
One proof is purely algebraic and uses the Pfaffian (as a matter of fact, the same result is true when one replaces ${\mathbb R}$ by another field).
The other proof is more familiar. Using the polar decomposition, one proves that ${\bf Sp}_{2n}({\mathbb R})$ is diffeomorphic to ${\mathbb R}^\ell\times{\bf U}_n$. Because the unitary group is connected, one obtains the connectedness of the symplectic group. We conclude with the fact that the determinant can take only the dicrete values $\pm1$.