# Answer to Question #178664 in Real Analysis for Chanchal

Question #178664
• Let(a↓n) ↓n€B be any sequence. Show that lim↓n-->♾️ a↓n =Liff for every E>0, there exists some N€N such that n>=N implies a↓n N↓e (L)
1
Expert's answer
2021-04-20T16:24:21-0400

Given

Let(a↓n) ↓n€B be any sequence. Show that lim↓n-->♾️ a↓n =Liff for every E>0, there exists some N€N such that n>=N implies a↓n N↓e (L)

By this we observed

To prove

Solution. Let > 0. By Theorem , note that

L < (Lx+ 1/2)1/x

and

L > (Lx- 1/2)1/x

there exists some N such that n ≥ N

We have anx< Lx+

and

anx> Lx-

This shows limn→∞ anx = Lx

.

Or

we can also solve by this way

Prove that {an} is a Cauchy sequence. Solution .

First we prove by induction on n that |an+1 − an| ≤ αn−1|a2 − a1| for all ...

an < A + ε. 2. ,

and there exists N2 such that for all n>N2,

bn < B + ε. 2 . But then if N = max(N1, N2),

then for any n>N we have an + bn < A + .

..xn − 1,n ∈ N.

Show that (xn) is decreasing and bounded below by 2. Find the ... 2 + xn,n ∈ N.

Show (xn) converges and find the limit. ...

2+2 = 4,

so xn < 2 for all n ∈ N by induction

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

#### Comments

No comments. Be first!

### Ask Your question

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS