I have that the symplectic group $Sp_{\mathbb{C}}(2n)$ is connected since we can construct a transitive action of $Sp_{\mathbb{C}}(2n)$ on the sphere $S^{4n-1}$ and the stabilizer subgroup $Sp_{\mathbb{C}}(2n-2)$ of the element $x_0 = (0,...,0,1)$. We know that $Sp_{\mathbb{C}}(2n)$ and $S^{4n-1}$ are compact and therefore it follows that if $Sp_{\mathbb{C}}(2n-2)$ is connected, then so is $Sp_{\mathbb{C}}(2n)$. We prove the claim that $Sp_{\mathbb{C}}(2n)$ is connected by induction.
My question is (besides if the reasoning above is correct) how to show $\textit{simple}$ connectedness.