<s>[INST] Proof by induction of a sum? Let $n ∈ N$. Prove by induction that there are $n$ ways to write the number $n$ as a sum $n=x_1+x_2+...+x_k$ where the $x_i$ are natural numbers and $x_1 ≤x_2 ≤...≤x_k ≤x_1+1$. 
For example, $5 = 5$, $5 = 1 + 1 + 1 + 1 + 1$, $5 = 1 + 1 + 1 + 2$, $5 = 1 + 2 + 2$, and $5 = 2 + 3$.
 [/INST] Here's a rough outline of a strategy: With $n=x_1+\cdots+x_k$, you can write $n+1=x_2+\cdots+x_k+(x_1+1)$. There is one allowed way to write $n+1$ that you can't get in this way, however.
Edit: To consider an example, here are the five ways to write $5$, with $x_1$ marked in red:
$5=\color{red}{5}$, $5=\color{red}{1}+1+1+1+1$, $\color{red}{1}+1+1+2$, $5=\color{red}{1}+2+2$, and $5=\color{red}{2}+3$
For each of these, take the red number, add $1$ to it and put it last, thus getting five ways to write $6$:
$6=\color{red}{6}$, $6=1+1+1+1+\color{red}{2}$, $6=1+1+2+\color{red}{2}$, $6=2+2+\color{red}{2}$, and $6=3+\color{red}{3}$.
Notice how this gives all the allowed ways to write $6$ as a sum, except for one, namely $6=1+1+1+1+1+1$. The same procedure works to get from $n$ to $n+1$ for all $n$, not just for $n=5$.
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