Where Can I Find Proofs for Limit Superior and Inferior Properties?

[Qyzren]

Hello, i just started in real analysis not too long ago. In my lecture notes There are 2 properties
lim sup (x+y) < & = lim sup x + lim sup y
lim sup (xy) < & = lim sup x * lim sup y
however there are no proofs for them in my notes. so i was wondering if anyone knew where a proof of these properties might lay or possibily give me a proof for these.
Thanks

 

[jostpuur]

Take a following lemma.

If $$a=\textrm{sup}\{a_1,a_2,a_3,...\}$$, then there exists a subsequence $$a_{k_1},a_{k_2},...$$ so that $$\lim_{i\to\infty} a_{k_i} = a$$.

Using this it is possible to conclude that

$$\textrm{sup}\{x_n+y_n, x_{n+1}+y_{n+1},...\} \leq \textrm{sup}\{x_n,x_{n+1},...\} + \textrm{sup}\{y_n,y_{n+1},...\}$$

and it's almost done.

EDIT: I used bad terminology. I guess for example $$a_1,a_1,a_1,...$$ isn't really a subsequence. Anyway, now that should be accepted as a "subsequence". The idea is then to take such a subsequence from the sequence of sums (on the left side of the inequality).