Suppose that $\lim x_n=L<\infty$. Then show there exists a continuous function $f:[0,1]\to \mathbb{R}$ where $f(1/n)=x_n$.
I'm not sure how exactly to go about this. What I thought about saying is that if we define $f(1/n)=x_n$ for small $n$ and $f(1/n)=f(0)$ if $n$ is large. Then I think this would make $f$ continuous but I'm a little unsure. Any help would be great. Thanks!