Let $U$ be an open subset in $\mathbb{R}^n$ and let $x\in U$ and a sequence $x_n\to x$
Prove that there exists a path $\gamma:(-\epsilon,\epsilon)\to U$ s.t:
$\gamma \ is\ C^2$
$\gamma(0)=x$
$\gamma'(0)\ne 0$
and $\gamma(-\delta,\delta)$ contains $x_n$ for infinitely many $n$ for every $0<\delta<\epsilon$
So I understand the last condition would come from the fact that $\gamma$ is continuous and that $x_n\to x$ because $\gamma(-\delta,\delta)$ is mapped to an open neighbourhood of x and that means that starting from some N, it will contain all of $x_n$ for $n>N$
The rest though, I really don't know how to prove..
Any help?