Hello. I want make sure im doing right step.

Question

Prove the supremum of the set A={2-1/n: n in natural number} is 3.

My answer
Proof: lets prove supA=2
Then 2 is an upper bound of A.
Let S'<2, then S' is not upper bound.
There exists x in A such that x> S'
Then 0<2-S'
There exists n in natural number such that 0<1/n <2-S'
Therefore supA=2. QED.

Answer 1

I mean prove the supremum of the set {2-1/n: n in N} is 2