Analysis Problem, limits & supremum, infimum and sequences

[retspool]

I have analysis quiz tomorrow and i am really poor at sequences.

I don't know where to begin

Let (sn) and (tn) be sequences in R. Assume that (sn) is bounded.

Prove that liminf(sn +tn)≥liminfsn +liminftn,

where we define −∞ + s = −∞ and +∞ + s = +∞ for any s ∈ R.

-thanks

 

[micromass]

Ok, what did you try already?

 

[retspool]

I got it,

I had to solve a problem before this one which gave me the result
lim infSn = -lim sup(-Sn)

And from a solved example i got

lim Sup(sn + tn) < lim supSn + lim SupTn

so multiplying b.s by (-1) and using -Sn instead for
we get

-lim Sup(-Sn - T) > lim Sup(-Sn) + lim Sup(-Tn)

There fore

lim Inf(Sn + Tn) > lim InfSn + lim InfTn

I had overlooked the solved example.