(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Fact:

Let a=lim sup a_{n}.

Then for all ε>0, there exists N such that if n≥N, then a_{n}<a+ε

Theorem 1:

If lim a_{n}= a exists, then lim sup a_{n}= lim inf a_{n}= a.

n->∞

Theorem 2:

If lim sup a_{n}= lim inf a_{n}= a, then

lim a_{n}exists and equals a.

n->∞

2. Relevant equations

N/A

3. The attempt at a solution

I was trying to see why theorems 1 & 2 are true.

How can we prove these rigorously?

I wrote down all the definitions, but still I don't know how to prove theorems 1 and 2.

Let a_{n}be a sequence of real numbers. Then by definition, a_{n}->a iff

for all ε>0, there exists an integer N such that n≥N => |a_{n}- a|< ε.

Also, lim sup a_{n}is defined as

lim sup{a_{n}: n≥N}

N->∞

(similarly for lim inf)

Any help is much appreciated! :)

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# Homework Help: Limit superior & limit inferior of a sequence

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