{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

solutionshw6-554-2010

# solutionshw6-554-2010 - n ≥ N Hence x n = n is both...

This preview shows page 1. Sign up to view the full content.

Solutions homework 6. Page 32 Problem 3. : | a + b | = | a | + | b | if and only if | a + b | 2 = ( | a | + | b | ) 2 if and only if ( a + b ) 2 = a 2 + 2 | a || b | + b 2 if and only if ab = | a || b | = | ab | . Now by deﬁnition x = | x | if and only if x 0 and thus ab = | ab | if and only if ab 0. Page 32 Problem 5. a. D ( a,b ) = 0 inf and only if | a - b | = 0 if and only if a = b . Also | a - b | ≥ 0 and 1 + | a - b | > 0 implies that D ( a,b ) 0. b. From | a - b | = | b - a | it follows that D ( a,b ) = D ( b,a ). c. From | a - c | ≤ | a - b | + | b - c | and problem 2.2: 3 it follows that D ( a,c ) D ( a,b ) + D ( b,c ). Page 34 Problem 1. Assume | x n | ≤ K for all n . Then | cx n | ≤ | c | K for all n , so ( cx n ) bounded. Asume also that | y n | ≤ L for all n . Then | x n - y n | ≤ | x n | + | y n | ≤ K + L , so ( x n - y n ) is bounded. Page 34 Problem 2. False, take y n = n and x n = 1 n . Then ( x n ) bounded, ( x n y n ) = (1 , 1 , ··· ) is bounded, but ( y n ) is not bounded. Page 36 Problem 1. Given N , we can take n = 2 N to get an even integer n N and n = 2 N + 1 to get an odd integer
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: n ≥ N . Hence x n = n is both frequently even and odd. This does not conﬂict with 3.2.4 as the ﬁrst case says via 3.2.4 that it is not true that x n = n is ultimately odd, while the second case says that it is not true that x n = n is ultimately even. Page 36 Problem 2. a. Note x n = 1 for n = 4 k + 1 with k ∈ Z . Hence given N take n = 4 N + 1 to get n ≥ N with x n = 1. b. Note ﬁrst that sin x ≤ x for all x ≥ 0 (this is used in Calculus 1 to show that (sin x ) = cos x ). Now 2 π n < 1 for all n ≥ 7, so also sin( 2 π n ) < 1 for all n ≥ 7. 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online