Prove that $x^n>1$ using induction given that $x>1$. Here was my way of (maybe) proving it: Step 1: $x^1>1$ which is true going back to what is given. Step 2: Assume $x^k>1$ Step 3: Prove $x^{k+1}>1$ (If someone could show me how to properly format exponents with operations in them please do so).
$x^{k+1}=x\cdot x^k>1 \cdot k$
$x \cdot x^k>1 \cdot k \rightarrow x^k>1$ So because I got back to the fact that $x^k>1$, and showed that $x^{k+1}$ is equal to it, am I done?
This is not right. First, what is the $k$ doing in the $1 * k$ term?
Second, and much, much, more importantly, you are going the wrong way around. You are effectively trying to go from $x^{k+1}>1$ to $x^k >1$, but you need to go just the other way around: you need to show that if $x^k>1$, then $x^{k+1}>1$
This is a very common mistake that people make with induction proofs. They start with what needs to be shown, and then show that what follows from that is something is true. Often the proof ends with some trivially true statement like $1=1$ or, as in your case, the inductive hypothesis.
However, notice that showing this shows absolutely nothing: of course $1=1$! So? Does the fact that some statement $P$ implies $1=1$ mean that statement $P$ is true? No, because you can derive $1=1$ from any statement, including false ones.
Likewise, if you assume that $x^k>1$, and then show that $x^{k+1}>1$ implies that $x^k>1$, you havent't shown that $x^{k+1}>1$ at all: once you assumed $x^k>1$, any statement implies $x^k>1$, and so in fact nothing can be concluded about the truth of $x^{k+1}>1$