Since lim_{n → ∞} s_n =L we must have:

Let c be a positive constant, and let ε>0.

Since lim_{n → ∞} s_n =L, we must have: for any t > 0, there exists N* such that for all n>N*, d(s_n,L) < t. So in particular, if we let t = εc, then we have: for all n > N*, d(s_n,L) < t = εc.

So we are done.

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The key here (imo) is to get the order of quantifiers right. We're fixing c and ε, and then finding the appropriate N* (or N as you called it in the question). So once you've fixed c, the proof becomes a matter of recognizing that εc is just some error threshold that the definition of convergence guarantees you can fall within, as long as you take large enough terms.