Is it possible to connect every compact set?

Choose a sequence $\varepsilon_n\to 0$ and a $\varepsilon_n$-net $N_n$ for each $n$. Assume $N_0$ is a one-point set. For each point in $x\in N_k$ choose a closest point in $y\in N_{k-1}$ and connect $x$ to $y$ by a curve. Note that we can assume that diameter of the curve is at most $\delta_k$ for a fixed sequence $\delta_k\to 0$.

Consider the union $K'$ of all these curves with $K$; observe that $K'$ is compact and path connected.