If the limit LIM (an * BN) = 0, then the limit Liman = 0 or the limit limbn = 0, please give a counterexample to prove it wrong

If the limit LIM (an * BN) = 0, then the limit Liman = 0 or the limit limbn = 0, please give a counterexample to prove it wrong

Sequence an: 0,1,0,1,0,1
Sequence BN: 1,0,1,0,1,0
LIM (an * BN) = 0, but Lim an does not exist and lim BN does not exist

Does Liman exist, limbn does not exist and lim (an * BN) exists

an=1/n
bn=n

If the limits of {an} and {BN} do not exist, can we conclude that the limits of {an + BN} and {an • BN} must not exist? Why?

The answer is No~
For example:
an=(-1)^n
bn=(-1)^(n+1)
Then, an + BN = (- 1) ^ n + (- 1) ^ (n + 1) = 0
Therefore, Lim an + BN = 0
Meanwhile, an * BN = (- 1) ^ n * (- 1) ^ (n + 1) = - 1) ^ (2n + 1) = - 1
Therefore, Lim an * BN = - 1
But obviously, an and BN diverge
If you don't understand, you're welcome to ask

What is the result of multiplying infinity by infinity? How to analyze such questions?

Infinity and infinitesimal are not numbers. Their multiplication is meaningless unless you specifically define it. In many mathematical fields, there are some multiplication operations that define infinity and infinitesimal from different angles, but many are different. In order to understand your doubts, I will give a common definition in calculus

Can an infinitesimal number be multiplied by an infinitesimal number to get an infinitesimal number?

be on the cards
The result of multiplying infinitesimal by infinity is uncertain
It may be equal to infinity, infinitesimal, or a constant that is not zero

What are the properties of infinity and infinitesimal?

Definition of infinitesimal: a variable whose limit is zero is called infinitesimal
(1) Infinitesimal is a variable and cannot be confused with very small numbers;
(2) Zero is the only number that can be infinitesimal
Definition of Infinity: a variable whose absolute value increases infinitely is called infinity
(1) Infinity is a variable and cannot be confused with large numbers;
(2) Infinity is a special unbounded variable, but unbounded variable is not necessarily infinity
(3) The algebraic sum (product) of infinitely many infinitesimals is not necessarily infinitesimal;
Theorem in the same process, the reciprocal of infinity is infinitesimal; The reciprocal of an infinitesimal that is not always zero is infinity
one
-= Lim X - > 0 (x > 0) in Y, then y - > is positive infinity at this time
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same
one
-= - Lim X - > 0 (x > 0) in Y, then y - > negative infinity at this time
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