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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?

[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?

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