Question

If a sequence <| Sn> is convergent, then the sequence <Sn> is
(a) convergent
(b)not convergent
(c) may or may not be convergent
(d) none of these​

Answer 1

Answer:

a . convergent ..

Answer 2

Answer:

• A sequence (sn) converges to a real number s if ∀ > 0, ∃Ns.t. |sn − s| <

∀n ≥ N. Saying that |sn−s| < is the same as saying that s− < sn < s+.

• If (sn) converges to s then we say that s is the limit of (sn) and write

s = limn sn, or s = limn→∞ sn, or sn → s as n → ∞, or simply sn → s.

• If (sn) does not converge to any real number then we say that it diverges.

• A sequence (sn) is called bounded if the set {sn : n ∈ N} is a bounded set.

That is, there are numbers m and M such that m ≤ sn ≤ M for all n ∈ N.

This is the same as saying that {sn : n ∈ N} ⊂ [m, M]. It is easy to see

that this is equivalent to: there exists a number K ≥ 0 such that |sn| ≤ K

for all n ∈ N. (See the first lines of the last Section.)

Fact 1. Any convergent sequence is bounded.

Proof: Suppose that sn → s as n → ∞. Taking = 1 in the definition of

convergence gives that there exists a number N ∈ N such that |sn−s| < 1 whenever

n ≥ N. Thus

|sn| = |sn − s + s| ≤ |sn − s| + |s| < 1 + |s|

whenever n ≥ N. Now let M = max{|s1|, |s2|, · · · , |sN |, 1 +|s|}. We have |sn| ≤ M

if n = 1, 2, · · · , N, and |sn| ≤ M if n ≥ N. So (sn) is bounded.

• A sequence (an) is called nonnegative if an ≥ 0 for all n ∈ N. To say that

a nonnegative sequence converges to zero is simply to say that:

∀ > 0, ∃Ns.t. an < ∀n ≥ N.

Fact 2. If (sn) is a general sequence then:

limn

sn = s ⇐⇒ limn

(sn − s) = 0 ⇐⇒ limn

|sn − s| = 0.

That is, the sequence (sn) converges to s if and only if the nonnegative sequence

(|sn − s|) converges to 0.

Fact 3. If (an) and (bn) are nonnegative sequences, with limn an = 0 and

limn bn = 0, and if C ≥ 0, then

limn

an + bn = limn

Can = 0.

Also, if limn an = 0 and if (bn) is any bounded sequence, then limn anbn = 0.

Fact 4. If (sn) and (tn) are sequences with sn ≤ tn for every n ≥ 1. If

limn sn = s and limn tn = t, then s ≤ t.

Fact 5: The ‘squeezing’ or ‘pinching rule’. Suppose that (sn),(xn), and

(tn) are sequences with sn ≤ xn ≤ tn, for every n ≥ 1. If limn sn = s and