We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is:

If we have two self-adjoint elements $T, S$ in $A$ with the same spectrum(may be infinite), Now can we also find two self-adjoint elements in $A$ with the same finte spectrum approximated $T, S$ in norm( within the same $\epsilon>0$)?

Hope some help or suggestion, thanks!