Is there a metric on $\mathbb{R}^S$ such that convergence in the metric is equivalent to pointwise convergence on $S$?

Question

Here, $S$ is any metric space and $\mathbb{R}^S$ is the set of functions from $S$ to $\mathbb{R}$. Our lecturer left this as an open question. He did however show this was true in the case $S = \mathbb{N}$ by defining the metric $d(x, y) = \sum_{k=1}^{\infty}2^{-k}\min\{1, \lvert x_k - y_k \rvert\}$ for $x, y$ in $\mathbb{R}^\mathbb{N}$. I feel the answer might be no in the case $S = \mathbb{R}$ but I have no idea how to prove it.