- #1
Mr Davis 97
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Homework Statement
Suppose that pn → p and each pn lies in lim S. We claim that p ∈ lim S. Since
pn is a limit of S there is a sequence (pn,k)k∈N in S that converges to pn as k →∞.
Thus there exists qn = pn,k(n)∈ S such that
d(pn, qn) <1/n.
Then, as n→∞ we have
d(p, qn) ≤ d(p, pn) + d(pn, qn) → 0
which implies that qn → p, so p ∈ lim S, which completes the proof that lim S is a
closed set.
Homework Equations
The Attempt at a Solution
The only part of the proof I don't understand is the following:
Since
pn is a limit of S there is a sequence (pn,k)k∈N in S that converges to pn as k →∞.
Thus there exists qn = pn,k(n)∈ S such that
d(pn, qn) <1/n.
Where does the 1/n come from? Why can we be sure that such a qn exists?