Recall that N(s; ε) = {x: |x − s| < ε) is the neighborhood of s of radius s. Prove the following.
(a) sn → s iff for each ε > 0 there exists M ϵ ℕ such that n ≥ M implies that sn ϵ N(s; ε).
(b) sn → s iff for each ε > 0 all but finitely many sn are in N(s; ε).
(c) sn → s iff, given any open set U with s ϵ U, all but finitely many sn are in U.
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