Question

IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In...


IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn:


i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn:
i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
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Sol: Here, given that,

In > 0 and yn > 0.

This implies that the sequences { {x_n} } and {{y _n}} are bounded.

Also, given that,

lim In < and lim Yn < 0 100

When

  In = 0, yn = 0.   

we get,

lim (Inyn) = 0 = ( lim In).( lim yn) 70 7-0 210

Which follows the inequality.

Now, we will proof the inequality for

In > 0, yn > 0

Let,

  lim In = A> 0 and lim Yn = B > 0 70

Also, we know that, a real number A is the limit superior of a bounded sequence {{x_n}} if and only if for each  € > 0, there exists a positive integer m such that

  u<UA 3+1 > ur

So, here, for some € > 0, there exists positive integers mı and my such that

  Tu <UA +> uz

and

  Yn <B+ , Vn> my

Let, M = max(mı, m2).

Therefore, for all n>M , we get,

  In Yn < (A+ 2A

  => In Yn < AB +E+ 4AB

=> In.Yn < AB +6   

  => lim (In:yn) <AB+ 70

  => lim (In:yn) <( lim In).(lim yn) +8 70 2001

  => lim (2n-yn) < (lim In).(lim yn) 70 100   Eis arbitrary

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In = 0, yn = 0.

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