How to show that Cx is a commitment to a integer of length lm
You want to prove knowledge of an opening $(x,r)$ to a Pedersen commitment, such that $x$ belongs to the range $[0,2^{l_m}]$. Such a proof is called a range proofs, and many methods exist, using bit-decomposition (commit to $x$ bit by bit, prove that the sum of the $x_i\cdot 2^i$ is indeed $x$, and prove that each committed value is a bit) or square decomposition techniques over hidden order groups. I discussed some of these methods in several answers, see here and here. The latest state-of-the-art method for range proofs is Bulletproof, which achieves relatively impressive efficiency guarantees, to the point that it's actually used in several real-world applications (e.g. the cryptocurrencies Monero and Mimblewimble, among others).