Here's a nice false proof of the continuum hypothesis.
Consider the rational numbers $\mathbb{Q}$ as a totally ordered field. We can add an indeterminate $T_0$ and make it positive but infinitely small (i.e., smaller than positive any element of $\mathbb{Q}$), that is, order $\mathbb{Q}(T_0)$ by lexicographic order of the Laurent series expansion at $0$. Then we can add another indeterminate $T_1$ and make it positive but infinitely small (i.e., smaller than any positive element of $\mathbb{Q}(T_0)$). This process can be iterated transfinitely and we can add $\aleph_1$ indeterminates $T_\iota$ for $\iota<\omega_1$, each infinitely smaller than all the previous ones. The resulting field $K = \mathbb{Q}(T_\iota)$ has cardinality $\aleph_1$ as is easy to show. Now any positive sequence converging to $0$ in $K$ must be eventually constant because it has to cross uncountably many $T_\iota$. So any Cauchy sequence in $K$ is eventually constant. So any Cauchy sequence in $K$ is convergent. So $K$ is complete. But since $K$ contains $\mathbb{Q}$, it contains $\mathbb{R}$. So we have a set of cardinality $\aleph_1$ containing $\mathbb{R}$, which proves the continuum hypothesis.
(The error, of course, is simply that the notion of "completeness" is wrong and its use is nonsense. But if you tell it quickly enough, many people will fall for it.)