Tangent bundle of submanifold

Hello,

I'm trying to do this problem

"Prove that if $\displaystyle M$ is a submanifold of a manifold $\displaystyle N$, then $\displaystyle TM$ is a submanifold of $\displaystyle TN$"

what I've thought so far is following:

Let $\displaystyle (U,k)$ be charts of $\displaystyle N$. Since $\displaystyle M$ is a submanifold of $\displaystyle N$, $\displaystyle M$ can be covered by a collection of $\displaystyle (U,k)$ with the property $\displaystyle M\cap U=k^{-1}(R^m\times {0})$ ($\displaystyle m$ is the dimension of $\displaystyle M$). Let $\displaystyle p\in M$, I think I have to prove that $\displaystyle T_p M$ can be covered by the collection of $\displaystyle T_p U$ with the property $\displaystyle T_pM\cap T_pU=Dk^{-1}(R^m\times {0})$

Could anyone give me suggestions or comments?