I am trying to align a series of inequalites/equalities in a proof using induction, using either {align*} or {IEEEeqnarray*}. I've followed the examples I've been able to find to the best of my ability, as well as looked at several short Latex tutorials and the Not So Short Introduction to Latex 2e. I am very new to Latex and am using Overleaf.
Here is my attempt using {align*}:
\begin{proof} Base case: If \(n=1\), then we have \(1 < 2^1 \Leftrightarrow 1<2.\) Thus, the theorem holds when \(n=1.\) \\ \\ Inductive hypothesis: Suppose the theorem holds for all values of \(n\) up to some \(k\), \(k \geq 1\). \\ Inductive step: Let \(n=k+1\). Then our left side is \begin{align*} k + 1 &\leq k + k && \text{since} k \geq 1. \\ &< 2^k + 2^k && \text{ since k<2^k by the inductive hypothesis.} \\ &= 2 \cdot 2^k \\ &= 2^{k+1}. \\ \end{align*} which is our right side. So, the theorem holds for \(n=k+1\). By the principle of mathematical induction, the theorem holds for all \(n \in \mathbb{N}\). \end{proof} This produces in Overleaf:
This code produces lines out of order and the text at the end of one of the inequalities appears at the end of the text of a subsequent line.
Here is my second attempt using {IEEEeqnarray*}, stripping down the argument to just the series of inequalities/equalities:
\begin{proof} \begin{IEEEeqnarray*}{rCl} k + 1 & \leq & k + k & \text{since} k \geq 1. \\ & < & 2^k + 2^k & \text{since} k < 2^k \text{by the inductive hypothesis.} \\ & = & 2 \cdot 2^k \\ & = & 2^{k+1}. \\ \end{IEEEeqnarray*} \end{proof} This produces in Overleaf:
The issue with this is the spacing of since $k \geq 1$ and for some reason "<" is not showing up properly in the explanation of the second line.
I've been spending a lot of time on this so any and all help is much appreciated. Thanks!
