Suppose $X$ is a smooth, compact, connected $n$-manifold without boundary which admits an immersion to $S^n$. Show that if $n>1$, then this immersion is a diffeomorphism.
Thanks for the very inspiring mentors, here I got some thoughts
$df_x$ is bijective. Because the tangent planes of the domain has the same dimension as domain; the tangent space of the codomain has the same dimension as codomain. But dim$X = n$, dim$S^n = n$, so $df_x$ maps from dim$n$ to dim$n$. Given immersion, $df_x$ is injective therefore bijective.
On the other hand, $f$ being an immersion told at that $df_x$ is nonsigular, hence a local diffeomorphism. I got stuck extending local diffeomorphism to global diffeomorphism. Is there a general strategy to achieve this(when this is true)?