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Lim inf/sup innequality question

  • #1
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x_n and y_n are bounded
[tex]
lim sup x_n+lim sup y_n>=lim x_r_n+lim y_r_n

[/tex]
this is true because there presented sub sequences converge to the upper bound of x_n and y_n .
the sum of limits is the limit of sums so we get one limit
and the of two convergent sequence is one convergent sequence
[tex]
lim(x_r_n+y_r_n)=limsup(x_n+y_n)
[/tex]
this is true because the sequence is constructed from a convergent to the sup sub sequences
so they equal the lim sup of x_n+y_n

now i need to prove that
[tex]
limsup(x_n+y_n)=>liminf x_n+limsup y_n
[/tex]

i tried:
[tex]
limsup(x_n+y_n)=limsup x_n+limsup y_n=>liminf x_n+limsup y_n
[/tex]
lim sup i always bigger then lim inf

so its true

did i solved it correctly?
 
Last edited:

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