It is obvious that $\mathbb{F}((X)) \subset F(R)$ (because any Laurent series is a quotient of a formal series by a power of $x$). So it remains to show that if $a = \sum_{n = 0}^{\infty} a_i x^i \in \mathbb{F}[[X]]$ is non zero, then the inverse of $a$ is in $\mathbb{F}((X))$.
If you assume $a_0 \ne 0$, check that $a$ has its inverse in $\mathbb{F}[[X]]$ (write $\left(\sum_{i = 0}^{\infty} a_i x^i \right) \times \left(\sum_{n = 0}^{\infty} b_i x^i\right) = 1$ and recursively compute the coefficients $b_i$).
In the general case, denote $n \ge 0$ the lowest integer such $a_n \ne 0$, then we can write $a = x^n b$ with $b = \sum_{i = 0}^{\infty} a_{i+n} x^i$. We know from the previous paragraph that the inverse of $b$ is in $\mathbb{F}[[X]]$, and the inverse of $x^n$ is in $\mathbb{F}((X))$, so the inverse of $a$ is also in $\mathbb{F}((X))$.