The Limit, supremum and infimum of a set.

Question

1. Guess the supremum and infimum of the set $A=\left\{\left(\frac{n+1}{2n}\right) \left(1+\frac{1}{n}\right)|n\in\mathbb{N}\right\}$ and prove by definition that they are indeed.

2. Prove by definition that the $$\lim_{n\to \infty}~~ \left(\frac{n+1}{2n}\right)\left(1+\frac{1}{n} \right)=\frac{1}{2}$$

Attempt:

1. My guess for $\inf$ and $\sup$ are $\inf(A)= 1/2$, $\sup(A)=2$. Although I'm not sure how to prove this by definition... Must I prove this by showing they are lower bounds and upper bounds respectively and then show that they are the smallest/greatest upper bound ? How would I go about doing this ?

2. I am quite certain that this isn't a valid proof since I haven't proved anything, it just works... but:

Let $A_n$ denote the $n^{th}$ term in the set A and consider $N=\frac{1}{\epsilon}$.

Then $\forall\epsilon>0,~~\left|A_n-\frac{1}{2}\right|<\epsilon~~~\forall n>N$

How could I make this more formal?

Answer 1

Hint: Both $1+\frac1n$ and $\frac{n+1}{2n}$ are decreasing.