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I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 3: Metrics and Norms ... ...

I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows:

View attachment 9201

I have not been able to make much progress ...

We have ...\(\displaystyle B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}\)

... and ...

\(\displaystyle B_r(0) = \{ y \in M \ : \ d(0, y) \lt r \}\)... and ...\(\displaystyle x + B_r(0) = x + \{ y \in M \ : \ d(0, y) \lt r \}\)But ... how do we formally proceed from here ...

Hope that someone can help ...

Peter===================================================================================

The above post refers to/involves the notion of an open ball ... so I am providing Carothers' definition of the same ... as follows:

View attachment 9202

The above post also refers to/involves the notion of a normed vector space ... so I am providing Carothers' definition of the same ... as follows:

View attachment 9203

View attachment 9204Hope that helps ...

Peter

I am focused on Chapter 3: Metrics and Norms ... ...

I need help Exercise 32 on page 46 ... ... Exercise 32 reads as follows:

View attachment 9201

I have not been able to make much progress ...

We have ...\(\displaystyle B_r(x) = \{ y \in M \ : \ d(x, y) \lt r \}\)

... and ...

\(\displaystyle B_r(0) = \{ y \in M \ : \ d(0, y) \lt r \}\)... and ...\(\displaystyle x + B_r(0) = x + \{ y \in M \ : \ d(0, y) \lt r \}\)But ... how do we formally proceed from here ...

Hope that someone can help ...

Peter===================================================================================

The above post refers to/involves the notion of an open ball ... so I am providing Carothers' definition of the same ... as follows:

View attachment 9202

The above post also refers to/involves the notion of a normed vector space ... so I am providing Carothers' definition of the same ... as follows:

View attachment 9203

View attachment 9204Hope that helps ...

Peter