How can I prove that $n < 2(n + 1)$? How can I logically prove that $n < 2(n + 1)$ assuming $n\in\mathbb{N}$?
This is probably a silly question but I am very tired.

Since $n$ is a natural number,
$$0 \le n$$
And we also have
$$0 < 1 + 1$$
So adding the two inequalities
$$0 < n + (1+1)$$
Adding $n$ to both sides,
$$n < n + (n + (1 + 1))$$
By associativity and commutativity,
$$n < (n+1)+(n+1)$$
By reversing the distributive rule,
$$n < (1+1)(n+1)$$
And simplifying,
$$n < 2(n+1).$$
