let E>0 be given
=>
there exists N1, N2 such that
|an -am | < E/2 for n,m >N1
|bn-bm | N2
=>
|an-am| + |bn-bm| N = max{N1,N2}
=>
|an+bn -am-bm| <=|an-am| + |bn-bm| N = max{N1,N2}
=>
|(an+bn) -(am+bm)| N
=>
an+bn is cauchy sequence
thus proved
an, bn are cauchy sequences => an, bn are bounded
let |an| <=A, |bn| <=B
there exists n1 n2 such that
|an-am| < E/2B for n,m >n1, => |an-am||bn| n1 => |anbn -ambn| n1
|bn-bm| < E/2A for n,m >n2 => |bn-bm||am| n2 => |bnam -ambm| n2
=>
|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| <|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| < E for n,m > N = max{n1,n2}
=>
anbn is a cauchy sequence
thus proved
Solution
let E>0 be given
=>
there exists N1, N2 such that
|an -am | < E/2 for n,m >N1
|bn-bm | N2
=>
|an-am| + |bn-bm| N = max{N1,N2}
=>
|an+bn -am-bm| <=|an-am| + |bn-bm| N = max{N1,N2}
=>
|(an+bn) -(am+bm)| N
=>
an+bn is cauchy sequence
thus proved
an, bn are cauchy sequences => an, bn are bounded
let |an| <=A, |bn| <=B
there exists n1 n2 such that
|an-am| < E/2B for n,m >n1, => |an-am||bn| n1 => |anbn -ambn| n1
|bn-bm| < E/2A for n,m >n2 => |bn-bm||am| n2 => |bnam -ambm| n2
=>
|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| <|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| < E for n,m > N = max{n1,n2}
=>
anbn is a cauchy sequence
thus proved.
ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH FORM 50 CÂU TRẮC NGHI...
let E0 be given=there exists N1, N2 such thatan -am E2.pdf
1. let E>0 be given
=>
there exists N1, N2 such that
|an -am | < E/2 for n,m >N1
|bn-bm | N2
=>
|an-am| + |bn-bm| N = max{N1,N2}
=>
|an+bn -am-bm| <=|an-am| + |bn-bm| N = max{N1,N2}
=>
|(an+bn) -(am+bm)| N
=>
an+bn is cauchy sequence
thus proved
an, bn are cauchy sequences => an, bn are bounded
let |an| <=A, |bn| <=B
there exists n1 n2 such that
|an-am| < E/2B for n,m >n1, => |an-am||bn| n1 => |anbn -ambn| n1
|bn-bm| < E/2A for n,m >n2 => |bn-bm||am| n2 => |bnam -ambm| n2
=>
|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| <|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| < E for n,m > N = max{n1,n2}
=>
anbn is a cauchy sequence
thus proved
Solution
let E>0 be given
=>
there exists N1, N2 such that
|an -am | < E/2 for n,m >N1
2. |bn-bm | N2
=>
|an-am| + |bn-bm| N = max{N1,N2}
=>
|an+bn -am-bm| <=|an-am| + |bn-bm| N = max{N1,N2}
=>
|(an+bn) -(am+bm)| N
=>
an+bn is cauchy sequence
thus proved
an, bn are cauchy sequences => an, bn are bounded
let |an| <=A, |bn| <=B
there exists n1 n2 such that
|an-am| < E/2B for n,m >n1, => |an-am||bn| n1 => |anbn -ambn| n1
|bn-bm| < E/2A for n,m >n2 => |bn-bm||am| n2 => |bnam -ambm| n2
=>
|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| <|anbn -ambn| + |bnam -ambm| < E for n,m > N = max{n1,n2}
=>
|anbn-ambm| < E for n,m > N = max{n1,n2}
=>
anbn is a cauchy sequence
thus proved
