Let x = inf E. Prove that x is an element of the boundary of E....

Let x = inf E. Prove that x is an element of the boundary of E. Here R is regarded as a metric space with d(p,q) = |q−p| for p,q ∈R

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Expert Solution

Defination : A point is said to be an boundary point of if every neighbourhood of contains a point of as well as a point of

Given ,

We will prove by contradiction that is an boundary point of

If is not an boundary point of then every neighbourhood of contains a point of as well as a point of ia not true .

There exist a neighbourhood of say such that

As is a neighbourhood of so there exist a such that

Now as and

which is a contradiction to because if then all other elements of should be greater than or equals to x .

So our assumption is not an boundary point of is wrong .

Hence is an boundary point of .

If you have any doubt or need more clarification at any step please comment .

Colby Messinger answered 1 year ago

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