Sequence that approximates union of N periodic sequences

Question

I have N monotonic sequences in the form of tn[i] = Tn*i + Pn (Tn > 0, Pn >=0), e.g.:

t1[i] = T1*i + P1
t2[j] = T2*j + P2
...

I need to find another sequence s[k] (not necessary in the same form) that approximates union of given sequences in such a way that:

1. For any sequence from the union tn and for any i there's such k that |s[k] - tn[i]| < E, where E is a given positive constant.
2. For any k following holds s[k] - s[k-1] > D, where D is a given positive constant.

How can I approach this class of problems? Engineer here, sorry if my terminology is off or if I didn't formalize the problem up to standards. I'm not even sure how to tag it.

The ultimate goal is to find minimal E (if it exists) for a given D (or for a given E find largest possible D).

Below is example for two sequences t1[j] = 5j and t2[i] = 6i with E=1.5 and D=4.5 (I don't think that's the optimal solution, but just to illustrate what I'm after).