# s7 - Math 5615H Introduction to Analysis I Fall 2011...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #7. Problems and Solutions. You can use the following Theorems which were discussed in class. Theorem 1. A subset K of a metric space ( X,d ) is compact in X if and only if every infinite subset E ⊆ K has a limit point in K . Theorem 2. A subset K of a metric space ( X,d ) is compact in X if and only if every sequence { p n } ⊆ K contains a convergent subsequence p n j → p ∈ K as j → ∞ . #1. Let K and F be two disjoint subsets of a metric space ( X,d ). Suppose that K is compact in X , and F is closed in X . Show that there is a constant c > 0, such that d ( x,y ) ≥ c > 0 for all x ∈ K and y ∈ F. Proof. Suppose that the given statement is not true. Then there are sequences { x n } ⊆ K and { y n } ⊆ F such that d ( x n ,y n ) < 1 n for all n = 1 , 2 , 3 ,.... Since K is compact, there is a convergent subsequence x n j → x * ∈ K as j → ∞ . We also have d ( y n j ,x * ) ≤ d ( x n j ,y n j ) + d ( x n j ,x * ) < 1 n j + d ( x n j ,x * ) → 0 as j → ∞ . This implies that x * is a limit point of { y n } ⊆ F , so that x * ∈ F , because F is closed. Then x * ∈ K ∩ F , in contradiction to our assumption that K and F are disjoint. Therefore, the given statement is true....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

s7 - Math 5615H Introduction to Analysis I Fall 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online