Upper Bound of Sets and Sequences: Analyzing Logic

[CGandC]

Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$
Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$
Suppose we look at the set $ A = \{ a_n | n \in \mathbb{N} \} $ .

I've been pondering for a while about the following 2 questions related to mathematical-writing , logic and set builder notation:

Questions:
1. How do we get using logic from the definition of upper bound for sets to the definition of the upper bound for sequences using the set $ A = \{ a_n | n \in \mathbb{N} \} $? My reasoning:
$ \forall \alpha \in A . \alpha \leq M \iff \forall \alpha.( \alpha\in A \rightarrow \alpha \leq M ) \iff $ $\forall \alpha.( \alpha \in A \rightarrow \exists n \in \mathbb{N}. \alpha = a_n \rightarrow \alpha \leq M ) \iff $ $ \forall \alpha.( \alpha \in A \rightarrow \exists n \in \mathbb{N}. \alpha = a_n \rightarrow \alpha \leq M \rightarrow a_n \leq M) $ Hence $ \forall \alpha \in A \exists n \in \mathbb{N}.( \alpha=a_n \land a_n\leq M $ ), now, this is not equivalent to the definition of upper bound of sequences above ( $ \forall n \in \mathbb{N}. a_n \leq M$ ), why? can you please give correct transitions? I think I made mistakes but It's confusing me to see how to write them correctly.

2. Someone told me that since every element of $ A = \{ a_n | n \in \mathbb{N} \} $ is generated by every element of $ n \in \mathbb{N} $ so therefore $ \forall \alpha\in A. \alpha \leq M$ is equivalent to $ \forall n \in \mathbb{N}. a_n \leq M$ .
How is that possible that the two statements are equivalent? Since for arbitrary $ \alpha \in C $ there exists a specific $ n \in N $ ( not arbitrary ) therefore it appears false to write $ \forall n \in \mathbb{N}. a_n \leq M$ but seems more reasonable to write $ \exists n \in \mathbb{N}. a_n \leq M$ .

 

[solakis1]

The complete and correct formalization of the above statatments are the following :

1)$M\in R$ is an upper bound of $A\subseteq R$ iff (and not if) $\forall a(a\in A\Rightarrow a\leq M)$

2)$M\in R$ is an upper bound of the sequence $(a_n)$ iff $\forall n ( a_n\leq M)$

OR

1)A, $A\subseteq R$ is bounded from above iff ) $\forall a((a\in A\Rightarrow\exists M(M\in R\wedge( a\leq M))$

2) $(a_n)\subseteq R$ is bounded from above iff $\forall n\exists M (M\in R\wedge (a_n\leq M)$

So depending on the words of the definition there different ways to formalise the definition

Also you do not use logic for formalizing a mathematical statement

To prove that two statement are equivalent you have to use logic

Now according to the words expressing that A has an upper bound M you can use the appropriate formalization

 

[CGandC]

Thanks, I understand now.

 

[solakis1]

I must also point out that only complet and correct formalization will allow you to write a correct formal proof