Why does frequency equal k/K where k=0,..., K-1 in Discrete Fourier Transform?
The DFT is, essentially, a sampled DTFT where the number of samples is \$K\$ and, thus, the spectrum is sampled at frequencies with spacing proportional to \$\frac{1}{K}\$ or, as in the image below, \$\frac{1}{N}\$



Note the \$k\$ indexes the samples in frequency just as \$n\$ indexes the samples in time.