# Boundedness of cell states in MC-LSTM (mass-conserving LSTM)

I'm currently reading Hoedt et al's paper on mass-conserving LSTM. In the corollary it is stated that

"[T]he memory cells, $$c_k^\tau$$, are bounded by the sum of mass inputs $$\sum_{t=1}^\tau x^t+m_c^0$$, that is $$\vert c_k^\tau \vert \leq \sum_{t=1}^\tau x^t+m_c^0$$. Furthermore, if the series of mass input converges, $$\lim_{\tau \rightarrow \infty} \sum_{t=1}^\tau x^\tau=m_x^\infty$$ , then also the sum of cell states converges."

I'm having trouble with the statement on convergence. Firstly, shouldn't the "exponent" in the sum be $$t$$ instead of $$\tau$$? I'm not even really sure what the index in the "sum of cell states" should be and which parameter is supposed to go to infinity.
Needless to say, I haven't been able to prove this statement, but I suspect it should be simple. The appendix says that the convergence follows with the comparison test.
I'd be grateful for any help on this matter!