<s>[INST] Use mathematical induction to prove that for all integers $n \geq 3,\, 2n + 1 < 2^{n}$ This is what I've got so far.
Let $P(n)$ be the statement that $2n + 1 < 2^n$
Basis: 
Let $n = 3$. Show that $P(3)$ is true.
$2(3) + 1 = 7$ and $2^3 = 8$.
Since $7 < 8$, $P(3)$ is true.
Inductive Hypothesis:
Suppose that $P(k)$ is true for an arbitrary integer $k\ge 3$.
   This implies that $2k + 1 < 2^k$.  
Show that $2(k + 1) + 1 < 2^{k + 1}$.
\begin{align*}
   2(k + 1) + 1 &= 2k + 2 + 1\\
                &= (2k + 1) + 2
\end{align*}
From the IH,
\begin{align*}
   2k + 1 < 2^k &\Rightarrow (2k + 1) + 2 < 2^k + 2\\
   &\Rightarrow (2k + 1) + 2 < 2^k + 2 \\ 
   &\Rightarrow (2k + 2) + 1 < 2^k + 2 \\ 
   &\Rightarrow 2(k + 1) + 1 < 2^k + 2
\end{align*}
Now we need to show that $2^k + 2 < 2^{k + 1}$.  
$2^k < 2^{k + 1}\Rightarrow 2^k + 2 < 2^{k + 1} + 2$ 
This is where I'm stuck. Any help on how to proceed is much appreciated.
 [/INST] Continuing from your step $2(k+1)+1<2^{k}+2$.
Since $k\geq 3$, then $2<2^{k}$.
So, $2(k+1)+1<2^{k}+2<2^{k}+2^{k}=2^{k+1}$
$\therefore 2(k+1)+1<2^{k+1}$.
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