Exercise 7Need parts a b and c Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x € X.a) For a sequence (n)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x)contains all the xn except maybe a finite number of them.b) In particular, show that x is a limit of (n)neN if and only if every open set U containing xcontains all the xn except maybe a finite number of them.c) Let (n)neN be a sequence in the metric space (X, d). Show that x = X is a limit of (n)neNif and only if for every e > 0 there exists ne EN such that d(xn, x) ≤ e, for all n ≥ ne. This isequivalent to d(xn, x) → 0 in R.

Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x = X. a) For a sequence (xn)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. h) In nautioular hon lima if 1 no m

Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x = X. a) For a sequence (xn)neN in X, show that x is a limit of (xn)neN if and only if every V = B(x) contains all the xn except maybe a finite number of them. h) In nautioular hon lima if 1 no m

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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