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I am struggling with this exercise:

Let $W$ be a $C^k$ manifold ($k > 0$) and $M,N \subset W$ be $C^k$ submanifolds such that for all $x \in M\cap N$, we have $T_xM \cap T_xN = 0$. Prove that $N \cap M$ is a discrete subset of $W$.

I was thinking about the inclusion $T_x(M\cap N) \subset T_xM \cap T_xN$ and, somehow trying to use Sard's Lemma. However I don't know how to put them together. Can someone help me? Thanks.

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    A definition of the notation $T_x$, or at least a phrase to explain its meaning, would be helpful to many Readers. Also the source of the exercise should be cited. – 2017-01-31
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    Much more basic than Sard's theorem: what would happen if you consider a sequence $x_n\to x$ of intersection points? – 2017-01-31
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    @hardmath $T_xM$ represents the tangent plane at the point $x$. – 2017-01-31
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    @JohnB could you please explain this idea more detailed? Thanks. – 2017-01-31
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    Since the origin of a tangent "plane" is relative (not absolute), it might be better to say $T_x M \cap T_x N = \{x\}$. – 2017-01-31
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    @hardmath By $0$ it was meant zero vector space: $T_x M$ and $x$ live in different objects. In $\mathbb{R}^n$ we tend to identify tangent space with some subset of $\mathbb{R}^n$ but generally it isn't so, as far as I remember. – 2017-01-31
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    @hardmath I was thinking about maybe proving that $M\cap N$ is a zero dimensional manifold, therefore a discrete set. The dimension of the manifold is as the same as the dimension of its tangent plane. Considering the inclusion I wrote, $T_x(M\cap N)$ has dimension zero and, then, $M\cap N$ has dimension zero. Am I wrong? – 2017-01-31
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    I suspect that is a sound approach, but there seem to be some substantial details to fill in. In general the intersection of two smooth submanifolds need not be a submanifold, – 2017-01-31
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    @Evgeny in this case $f$ and $g$ are charts for $N$ and $M$ respectively right? – 2017-01-31

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