Without loss of generality $X$ is affine, so embed it in projective space and apply the Bertini Theorem to conclude that $X$ contains a smooth affine curve $C^{\circ}$ missing $N$ points from its projective completion $C$. Such a guy will remain smooth modulo $v$ for almost all places of $v$ -- this follows immediately from the Jacobian condition -- and now you are reduced to Weil's theorem: you have a family of smooth affine curves $C^{\circ}$ over finite fields $\mathbb{F}_v$ of fixed genus and missing $N_v \leq N$ points from its projective completion (in fact $N_v = N$ for sufficiently large $v$). Now the Weil bounds tell you that the number of $\mathbb{F}_v$-rational points goes to infinity with $\# \mathbb{F}_v$. Finally, apply Hensel's Lemma.

Without loss of generality $X$ is affine, so embed it in projective space and apply the Bertini Theorem to conclude that $X$ contains a smooth, geometrically integral affine curve $C^{\circ}$ missing $N$ points from its projective completion $C$. Such a guy will remain smooth modulo $v$ for almost all places of $v$ -- this follows immediately from the Jacobian condition -- and now you are reduced to Weil's theorem: you have a family of smooth affine curves $C^{\circ}$ over finite fields $\mathbb{F}_v$ of fixed genus and missing $N_v \leq N$ points from its projective completion (in fact $N_v = N$ for sufficiently large $v$). Now the Weil bounds tell you that the number of $\mathbb{F}_v$-rational points goes to infinity with $\# \mathbb{F}_v$. Finally, apply Hensel's Lemma.