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).

 

[tacman]

The properties you mentioned are known as the limit superior and limit inferior (also known as the lim sup and lim inf) of a sequence. They are important concepts in real analysis, and it is great that you are already starting to learn about them.

To prove these properties, we first need to understand what the limit superior and limit inferior are. The limit superior of a sequence (x_n) is defined as the largest number that the sequence approaches as n tends to infinity. In other words, it is the supremum (or least upper bound) of all the numbers in the sequence. Similarly, the limit inferior is the infimum (or greatest lower bound) of the sequence.

Now, let's look at the first property: lim sup (x+y) < & = lim sup x + lim sup y. This property states that the limit superior of the sum of two sequences is less than or equal to the sum of their individual limit superiors. To prove this, we can use the definition of limit superior. Let's denote the limit superior of the sequence (x_n) as S and the limit superior of the sequence (y_n) as T. This means that for any ε > 0, there exists an N_1 such that for all n > N_1, we have x_n < S + ε. Similarly, for the sequence (y_n), there exists an N_2 such that for all n > N_2, we have y_n < T + ε.

Now, let's consider the sequence (x_n + y_n). For any ε > 0, we can choose N = max{N_1, N_2}. Then for all n > N, we have x_n + y_n < S + T + 2ε. This means that S + T + 2ε is an upper bound for the sequence (x_n + y_n). But since S is the supremum of the sequence (x_n), we know that x_n < S + ε for all n > N_1. Therefore, x_n + y_n < S + T + 2ε for all n > N_1. This shows that S + T + 2ε is also an upper bound for the sequence (x_n + y_n). Since ε is arbitrary, we can conclude that S + T is the supremum of the sequence (x_n + y_n), which is the definition of lim sup (