Let $x_n=\frac{1}{2}x_{n-1}+1$ where $x_1=1$. Show that the sequence $x_1, x_2, x_3,...$ is bounded by 2.
Now I know that the proof is simple through induction. What I want to know if the following argument is as valid, formally.
Proof:
Suppose $\exists n\in\Bbb{N}$ such that $x_n \leq 2$ where $x_n=\frac{1}{2}x_{n-1}+1$. Then $\frac{1}{2}x_n+1\leq2$ $\Rightarrow$ $\frac{1}{2}x_n-1\geq1$ $\Rightarrow$ $x_{n-1}\geq2$. Proceeding in this fashion, we will then end with $x_1\geq2$. But since $x_1=1$, we have reach a contradiction.
Is induction very slightly camouflaged. In induction you prove $$P(1)\hbox{ and }\forall n>1: P(n-1)\implies P(n).$$ You have proved $$P(1)\hbox{ and }\forall n>1: \neg P(n)\implies\neg P(n-1).$$ (the contrapositive of the implication)