Question

A convergent sequence in R can have what limit​

Answer 1

Step-by-step explanation:

A sequence which has a limit is said to be convergent. A sequence with no limit is called divergent. is convergent with limit 0.

Answer 2

Answer:

to prove “For all convergent sequences the limit is unique”.

The negation of this is “There exists at least one convergent sequence

which does not have a unique limit”.

This is what we assume.

On the basis of this assumption let{an}n∈N denote a sequence with more than one limit, two of which are labelled as l`1 and`2 with l`1 not= l `2.

Choose ε = 1/ 3 |l`1 − l `2| which is greater than zero since l`1 6= l`2.

Since l`1 is a limit of {an}n∈N we can apply the definition of limit with our choice of ε to find N1 ∈ N such that |an − l`1| < ε for all n ≥ N1.

Similarly, as l`2 is a limit of Hence for all convergent sequences the limit is unique.{an}n∈N we can apply the definition of limit with our choice of ε to find N2 ∈ N such that |an − l`2| < ε for all n ≥ N2.

Hence for all convergent sequences the

Hence for all convergent sequences the limit is unique.