Answered: Consider the space Z+ with the finite…

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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Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in Z+ given by xn = n+7. To what point or points does the sequence converge?

Expert Solution

Step 1

$(ℤ, +)$ with finite complement topology.

$(ℤ +, τ_{C})$

where $τ_{C} = {U \subseteq ℤ + | ℤ + ∖ U i s f i n i t e}$

The sequence $x_{n} = n + 7$ converges to all points in $ℤ +$ .

Take any $m \in ℤ +$

Any open set containing $m$ is of the form $U$ such that $ℤ + ∖ U$ is finite,then we have an open set in $ℤ +$ which does not take finitely many values of $ℤ +$ other than $m$ .

So, we have an $N \in ℕ$ such that $x_{n} \in U$ for $n ⩾ N$

So , by definition of a convergent of a sequence , $x_{n} \to m$

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