Restricting a smooth vector bundle to a submanifold

Question

[Vector Bundle Chart lemma]

The above is an example in Lee's smooth manifolds. My question is about the case where $S$ an immersed (or embedded) submanifold of $M$. By the chart lemma, $E|_S$ is indeed a vector bundle. Then is $E|_S$ an immersed (or embedded) submanifold of $E$?

In both cases, immersion of the inclusion map $i:E|_S \rightarrow E$ can be easily checked by coordinate charts in $E|_S$ and $E$. So, how to prove whether the topology of $E|_S$ is the subspace topology of $M$?