- #1
tamintl
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Is the following statement true or false: 'if (cn) and (dn) are bounded sequences of positive real numbers then:
lim sup (cndn) = (lim sup cn)(lim sup dn)
for all n in the positive reals. cn and dn are bounded.
Since cn and dn are bounded we know they converge.
Hence, by the theorem lim(cndn)=lim cnlim dn we can say that lim(cn)=lim sup (cn) and lim(dn)=lim sup (dn)
Hence, lim sup (cndn) = (lim sup cn)(lim sup dn)
HENCE, TRUE!
******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******
Homework Statement
Is the following statement true or false: 'if (cn) and (dn) are bounded sequences of positive real numbers then:
lim sup (cndn) = (lim sup cn)(lim sup dn)
Homework Equations
The Attempt at a Solution
for all n in the positive reals. cn and dn are bounded.
Since cn and dn are bounded we know they converge.
Hence, by the theorem lim(cndn)=lim cnlim dn we can say that lim(cn)=lim sup (cn) and lim(dn)=lim sup (dn)
Hence, lim sup (cndn) = (lim sup cn)(lim sup dn)
HENCE, TRUE!
******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******