2.(a) Suppose that𝑋(p, 𝑚) isnotlhc. Then for every neighborhood𝑆of (p, 𝑚),there exists (p′, 𝑚′)∈𝑆such that𝑋(p′, 𝑚′)∩𝑇=∅.In particular, forevery open ball𝐵𝑛(p, 𝑚), there exists a point (p𝑛, 𝑚𝑛)∈𝐵𝑛(p, 𝑚) suchthat𝑋(p𝑛, 𝑚𝑛)∩𝑇=∅. ((p𝑛, 𝑚𝑛)) is the required sequence.(b) By construction,∥p𝑛−p∥<1/𝑛→0 which implies that𝑝𝑛𝑖→𝑝𝑖for every𝑖. Therefore (Exercise 1.202)∑𝑝𝑛𝑖˜𝑥𝑖→∑𝑝𝑖˜𝑥𝑖< 𝑚and𝑚𝑛→𝑚and therefore there exists𝑁such that∑𝑝𝑁𝑖˜𝑥𝑖< 𝑚𝑁which implies that˜x∈𝑋(p𝑁, 𝑚𝑁)(c) Also by construction𝑋(p𝑁, 𝑚𝑁)∩𝑇=∅which implies𝑋(p𝑁, 𝑚𝑁)⊆𝑇𝑐and therefore˜x∈𝑋(p𝑛, 𝑚𝑛) =⇒˜x/∈𝑇The assumption that𝑋(p, 𝑚) is not lhc at (p, 𝑚) implies that ˜x/∈𝑇, contra-
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