Let $(M,\omega)$ a simply connected symplectic manifold. Suppose there are symplectomorphism $\phi_k: M\rightarrow M$ such that
1)each $\phi_k$ has unique fixed point in $M$.
2)$\phi_k \rightarrow \text{id}$ in the $C^0$-topology on the group $Symp(M)$ of symplectomorphism of $M$
Show that $M$ is not compact.
I find this question from https://www.dpmms.cam.ac.uk/study/III/2005-06/SymplecticTopology/STsheet4.pdf.
It is very easy if 2) is repleced by $C^1$ convergence. Because $M$ is simply connected and we could identify fixed points of $\phi_k$ with critical points of some functions on $M$ via the Weinstein's neighbourhood theorem.
But current condition is $C^0$ convergence. How to do it? I have no idea.