- #1
transgalactic
- 1,395
- 0
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?
Last edited: