The function $\exp: \mathfrak{su}(n)\to SU(n)$ is surjective
I am trying to prove this by induction and I have already showed for $n=1,2$. I have also proven that given $x\in SU(n)$, there is $y\in SU(n)$ such that $b=yxy^{-1}$ is a block matrix where each block is an element of $SU(1)$ or $SU(2)$, but I don't know how to use all this to prove that $\exp: \mathfrak{su}(n)\to SU(n)$ is surjective. Some help? Thank you.
Note the exponential map $\exp:\mathfrak{g}\to G$ is $G$-equivariant, where $\mathfrak{g}$ has the adjoint representation; that is $\exp(gXg^{-1})=g\exp(X)g^{-1}$ for all $g\in G$ (assuming $G$ is a matrix Lie group).
Therefore, it suffices to show every diagonal element of $SU(n)$ is in the range of $\exp$. This should be pretty straightforward, since these all look like $\mathrm{diag}(e^{i\theta_1},\cdots,e^{i\theta_n})$ such that $\theta_1+\cdots+\theta_n=2\pi n$ (and you can show that $n=0$ without loss of generality here).