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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 defintion 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 defintion 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$ .