Proving the Maximal Member in a Converging Sequence An > 0 to 0

[transgalactic]

An is a bounded sequence which converges to 0.
An>0 for every n.
prove that in this sequence we have a maximal member
??

again its obvious because if a sequence is descending and it converges from a positive number
to 0.
so 0 is the larger lower bound
and our first member must be the larger

how to transform it to math?

 

[VeeEight]

If it converges to 0, then finitely many members of the sequence take on a different value (that number may be small or large but it is still finite) and the rest of the terms approach 0.

 

[transgalactic]

how to prove it mathematically?

 

[VeeEight]

By combining what you and I wrote and using proper notation