A curve in Euclidean space $\mathbb{R}^n$, $n \geq 2$ is $analytic$ if the coordinates of its points $x= x_{1},...,x_{n}$ can be expressed as analytic functions of a real parameter $x_{i}=x_{i}(t)$, $i=1,...,n$ and $\alpha \leq t \leq \beta$ and the derivatives $x'(t_{0})$ do not simultaneously vanish for any $t_{0} \in [\alpha, \beta]$.
I search for a proof of the following fact:

If the set of intersection points of two analytic curves is infinite, then these curves coincide.

Can we prove the same as above if we relax the condition that the coordinates are analytic to the condition that the coordinates belong to the class $C^{\infty}$?

Edit: Thanks to below remark by Ramiro, to obtain the above implication, we have to assume that any two curves $K_{1}, K_{2}$ as in the question are such that $K_{1} \cap K_{2}$ is not another analytic curve.

2nd edit: As suggested Peter, we can reformulate our question as follows:
   Consider two immersed curves which are parameterized real analytically on compact intervals. If they have an infinite number of different intersection points, then their union is again a real analytic immersed curve. 


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