Interpolation and harmonics A real valued causal sequence $x1[n]$ exists with length of the sequence being $N$. 
Valid indices of x conform to $0 \le n \le N-1 $

The DFT of x[n] is:
$$
X1[k] =  \sum_{n=0}^{N-1} x1[n].e^{-j.2.\pi.k.n/N} 
$$

The normalized frequency spectrum exists from 0 to $2\pi$. This frequency spectrum contains discrete frequencies which are integer factors of $\frac{2 \pi}{N}$. 

A properly sampled signal fulfilling Nyquist criteria will have the valid range of normalized frequencies from $0$ through $(\frac{N}{2} - 1)$ $\frac{2\pi}{N} $.

As an example, for N=8, the valid frequencies are $0$, $2\pi.n/N$, $4\pi.n/N$ and $6\pi.n/N$.

When $x1[n]$ is upsampled with a factor of 2, we essentially insert a $0$ after each sample. This creates a new sequence $x2[n]$ with a length of $2N$.

The DFT of this new sequence will now be:
$$
X2[k] =  \sum_{n=0}^{2N-1} x2[n].e^{-j.\pi.k.n/N} 
$$

The discrete frequencies of this new sequence are $0$, $\pi.n/N$, $2\pi.n/N$, $3\pi.n/N$, $4\pi.n/N$, $5\pi.n/N$, $6\pi.n/N$, $7\pi.n/N$ and so on.

So it is clear that each new sample that was added to $x1[n]$ has introduced a new frequency component. 

I have two questions now:


As the harmonics (as the math seems to suggest) lies amongst the desired frequencies (e.g., $\pi.n/N$  is less than $2\pi.n/N$, and $3\pi.n/N$  is less than $4\pi.n/N$), shouldn't the interpolation filter be a comb filter?
When I take a FFT of $x2[n]$, I expect to see the harmonics in the original pass-band. But instead, the frequency spectrum of x1[n] has been replicated.


What have I misunderstood?
Upsampling requires two steps:


Inserting zeros. This does indeed replicate the spetcrum in the frequency domain. For N zeros inserted you get N copies of the original spectrum
Lowpass filtering to remove the mirror spectra. That's often also called an "interpolation filter".


The choice of interpolation filter depends a lot on the requirements of your specific application (signal to noise ratio, spectral suppression, shape of original spectrum, transient behavior, phase distortion, latency, MIPS, etc.). There is no "one size fits all" solution