Taking the lim operation on both sides

[transgalactic]

is it legal to take the limit on both sides of an expression

i particular

when i have

sup x_n <=sup y_n

is it ok to do

limsup x_n <=limsup y_n

is the limit of the supremums group of subsequences is the limsup on the sequence
??

 

[yyat]

One of the properties of the limit is that if $$a_n\le b_n$$ for all n, then $$\lim_{n\to\infty}a_n\le\lim_{n\to\infty}b_n$$ provided that these limits exist.
Does this answer your question?

 

[Office_Shredder]

This isn't quite that case yyat.

Counterexample: x_n = 1 if n=1, 0 otherwise
y_n = 1/2 for all n

supx_n = 1
supy_n = 1/2

limsup x_n = 0
limsup y_n = 1/2

 

[yyat]

This isn't quite that case yyat.

Counterexample: x_n = 1 if n=1, 0 otherwise
y_n = 1/2 for all n

supx_n = 1
supy_n = 1/2

limsup x_n = 0
limsup y_n = 1/2

I don't see how this contradicts anything I wrote.

But you are right, if transgalactic means that sup x_n=sup{x_n,n>=0}, then the statement is wrong by the counterexample you gave (with x_n, y_n switched). I assumed it meant sup{x_n,n>m}<=sup{y_n,n>m} for all m.