Let > 0 and > 0, and a R. Show that V (a) V (a) and V (a) V (a) are -neighborhoo

Question

Let > 0 and > 0, and a R. Show that V(a) V (a) and V(a) V (a) are -neighborhoods of a for appropriate values of .

Explanation / Answer

Proof: Given that >0, >0 and a in R. Given that V(a) V (a) and V(a) V (a). Claim: for proper , V(a) V (a)=V(a) and V(a) V (a)=V(a). Choose =minimum of{,}. The we can see that V(a) V (a)=V(a). Also choose =maximum of{,}. The we can see that V(a) V (a)=V(a). Since V(a) V (a)= and V(a) V (a) contains a, and both are open sets in R, then it is clear that V(a) and V(a) are open and contains a. Because union of two open set are open in R and intersection of two open sets in R are again open in R. Hence the proof.

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.