**Firstly, it is a misconception that tachyons do not exist in the naive (i.e. non-GSO, inconsistent) RNS superstring theory. It is just that, RNS superstring theory can go under a GSO truncation, leading to the Type IIB and Type IIA string theories, which do not have tachyons.**

In the Bosonic String theory, the mass spectrum of closed strings is given by (in naturaol units where $\ell_s=\hbar =c_0=1$:

$$m=\sqrt{N+\tilde N-2}$$

$N$ and $\tilde N$ can only take discrete values as non-negative, either half-integers, or integers. For example, if you set $N=\tilde N = 0$, which is clearly for the ground state $|0\rangle $. :

$$m=\sqrt{-2}=\sqrt2 i$$

I.e. an imaginary mass. Therefore, a tachyon. This is also for the open string sector, whose mass spectrum is $m=\sqrt{N-1}$, then, when $N=0$, at the ground state $|0\rangle $, i.e., it' is a tachyon.

**The same problem holds in the RNS string theory.**

If you analyse the mass spectrum of the RNS superstring theory, you see that the same problem holds.

$$m=\sqrt{N+\tilde N - A}$$

Where $A=0$ in the RR sector, $A=1$ in the N-SN-S sector, and $A=\frac{1}{2} $ in both the RN-S and N-SR sectors. Clearly, in all sectors; but the RR sector, there is a tachyonic ground state.

This initiates the need for the GSO Truncation/(Projection).