follows:

The optimization problem is feasible if $$\label{eq:SINR-bound}
\gamma < \frac{S (\alpha -1) (1-\epsilon^2) }{ 1 + \epsilon^2 S (\alpha -1)}.$$

Observe that $\underline {\mathrm{SINR}}$ in is a monotonically increasing function of $M$. Therefore, an upper limit can be computed by letting $M \rightarrow \infty$. This yields $$\underline {\mathrm{SINR}}
\xrightarrow{M \to \infty}
 \frac{(1-\epsilon^2)^2}{(1-\epsilon^2) \big( \frac{1}{\beta( \alpha-1)} + \epsilon^2 \big)},$$ which is an increasing function of the optimization variable $\beta$. Under the constraint $\beta K \leq S$, the maximal value of $\beta$ is $S$ and this value is obtained for $K=1$. Putting all these facts together and using simple algebra lead to .

The above lemma shows that the maximal SINR level is limited only by the hardware impairments, through $\epsilon$, and by the severity of the pilot contamination, which is determined by the pathloss exponent $\alpha$ and the coherence block length $S$. These are the only limiting factors as $M \rightarrow \infty$. Assume for example that $\epsilon = 0.05$ [@Bjornson2013c; @Bjornson2014a] and consider the relatively conservative propagation parameters $\alpha=3$ and $S = 200$. For these numbers, the upper limit of the average SE per UE is $\log_2(1+199.5) \approx 7.65$, which is substantially higher than the SE of contemporary systems [@Holma2011a]. 