What is the upper bound for n/(2n+1)​

You need a bit more. That the limit of the sequence is 2
2
isn't enough. You also need, that this sequence is monotonous. Then you can say that 2
2
is the upper bound.

You can try it this way:

As
n
approaches infinity, you see that
n
gets much bigger than 1, so you can neglect it and you get
lim→∞21+=lim→∞2=2.
lim
n
→
∞
2
n
1
+
n
=
lim
n
→
∞
2
n
n
=
2.

Because
a
n
is monotonous, you see, that this is indeed the upper bound.