If $M$ is a regular submanifold of $N$,then for any $x \in M$, use local coordinate,we can naturally take $T_xM$ as a subspace of $T_xN$, now I want to prove $TM$ is a submanifold of $TN$, where $TM,TN$ represent the tangent bundle.
Generally,for two manifold $A,B$, construct a map $f:A \rightarrow B$, such that $f_*:T_xA \rightarrow T_{f(x)}B$ is a monomorphism, thus $f$ is a immerse, then $A$ is a submanifold of $B$.
But I think consider tangent space of a tangent bundle is difficult, I hope someone can give me some hints about my problem. Thanks!