Suppose we have the following minimization problem:

\text{minimize}_{z \in \mathbb{R}^{N}} \sum_{i=1}^{N} \frac{|z_i|}{(|z_{n,i}|+ \varepsilon_n)^{1-q}} subject to A \bold{z} = \bold{y} . How do we prove the following:

Proposition. For any nonincreasing sequence (\varepsilon_n) of positive numbers and for any initial vector \bold{z}_0 satisfying A \bold{z}_0 = \bold{y} , the sequence above admits a convergent subsequence?