Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) = n-00 n-00

Q: Let {a;} be a sequence in R. Prove lim an = 0, if and only if, lim Jan| = 0. n-00 If lim Jan] = 1,…

A: Suppose limn→∞an=0 Then by definition of limit, for given ε>0,|an-0|<ε for n>1…

Q: Evaluate a* -b* lim - b* x→0 X

A:

Q: Prove the limit Statement lim (5x-8x-4) = 12

A:

Q: If a sequence C₁, C2, C3,... has limit K then the sequence e limit of the sequence given by b₁ = (n)…

A:

Q: Prove the limit statement 23) lim (5x - 1) = 19 X4

A: Substitute the variable with the value.

Q: 3. Let (Sn) be a sequence of nonnegative real numbers, and suppose lim s, = 0. Prove that lim S, =…

A:

Q: 3 show that if lin N-)P number . an = a, then lim 29 = 2a for all real 3-)

A: Given : limn→∞ an=a To prove : limn→∞  λan=λa.

Q: Compute 4n — бп? lim Vlan| for a,n %3D 3n2 +7 n-00

A:

Q: Let (an) and (bn) be two bounded sequences of positive real numbers. Prove that lim sup(anbn) (lim…

A: Given two bounded sequences of positive real numbers  and .We have to prove that, .To do this, use…

Q: 10. Suppose (an}, {bn} are sequences of nonnegative real numbers with him bn = b = 0 and lim a, = a.…

A: 10. Given that are sequence of nonnegative real numbers with        and       To prove that:

Q: 3* –1 lim V1+x -1,

A:

Q: Show that if lim supn→∞an = L for a finite real number L and > 0, then an> L+ ε for only finitely…

A: The objective is to prove the statement: If limsup an=Ln→∞ for a finite real number L and ε>0,…

Q: lim cob lo) -の

A: Given : limθ→-∞cosθ3θ

Q: Find the limit

A: limx→∞2x-1x+1

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.