(EDITED) Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$. Let $C_n = \{x \in [0,1]: \forall m > n, \;|f_n(x) - f_m(x)|\le 1/3\}$. Then $C_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C_N$ has nonempty interior. But this is impossible since $f_n$ is continuous and both $\Delta$ and its complement are dense.

(EDITED) Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$. Let $C_n = \{x \in [0,1]: \forall m > n, \;|f_n(x) - f_m(x)|\le 1/3\}$. Then $C_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C_n$ has nonempty interior. But this is impossible since $f_n$ is continuous and both $\Delta$ and its complement are dense.

Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$. Let $C_n = \{x \in [0,1]: |f_n(x) - \chi_\Delta(x)|\le 1/3\}$. Then $C_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C_N$ has nonempty interior. But this is impossible since $f_n$ is continuous.

(EDITED) Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f_n$ converges pointwise to $\chi_\Delta$. Let $C_n = \{x \in [0,1]: \forall m > n, \;|f_n(x) - f_m(x)|\le 1/3\}$. Then $C_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C_N$ has nonempty interior. But this is impossible since $f_n$ is continuous and both $\Delta$ and its complement are dense.