1. (a) Prove that x is an accumulation point of a set S iff there exists a sequence (s n ) of points in S \{x} such that (s n ) converges to x. (b) Prove that a set S is closed iff, whenever (s n ) is...

1. (a) Prove that x is an accumulation point of a set S iff there exists a sequence (s_n) of points in S \{x} such that (s_n) converges to x.

(b) Prove that a set S is closed iff, whenever (s_n) is a convergent sequence of points in S, it follows that lim s_n
is in S.

2. Recall that N(s; ε) = {x : | x – s | <>

(a) s_n
→ s iff for each ε > 0 there exists M ∈ N such that n ≥ M implies that s_n
∈ N(s; ε ).

(b) s_n
→ s iff for each ε > 0 all but finitely many s_n
are in N(s; ε ).

(c) sn → s iff, given any open set U with s ∈ U, all but finitely many s_n
are in U.