Comment by Gro-Tsen on Is every real number the limit of a sequence of irrational numbers, constructively?

@MohammadTahmasbizade The counterexample is easily modified to additionally satisfy $1\leq x\leq 2$, so it is not provable that a real in $[1,2]$ (or, a fortiori, a nonzero real) is the limit of a sequence of irrationals.