I'm learning about how use mathematical induction. I'm tasked with proving the inequality shown in (1). It is a requirement that I use mathematical induction for the proof.
$(1) \quad P(n):\quad 2n+1 < 2^{n}, \quad n \ge 3$
I would like some feedback regarding whether my proof is valid and if my use of the induction hypothesis is correct. My proof:
Prove base case:
$(2) \quad P(b):\quad 2\cdot3+1 < 2^{3}, \quad b=1$
$(3) \quad P(b):\quad 7<8$
Assume that $P(n)$ is true for an arbitrary $k$, with $n \ge k \ge 3$. (This is the induction hypothesis, IH)
$(4) \quad P(k):\quad 2k+1 < 2^{k}$
The inductive step:
$(5) \quad P(k+1):\quad 2(k+1)+1 < 2^{k+1}$
$(6) \quad 2k+3 < 2^{k}\cdot 2$
Using the IH we get (7). I just added 2 to both sides of the inequality so that the LHS in (6) and the IH is equal.
$(7) \quad 2k+3 \overset {IH}{<} 2^{k}+2$
Subtracting 2 from both sides yields us $P(k)$:
$(8) \quad 2k+1 < 2^{k}$
Which concludes the proof that $P(n)$ is true for every $n \ge 3$
Is my proof valid? Is my use of the IH correct?
I am aware that a similar problem is asked here.