Let $P(n)$ be a number of all integer partitions of n. Then, show that $P(1) + P(2) + ... P(n) < P(2n)$.
The solution from the textbook: If $\pi$ is a partition of $i$ for $i \leq n$, then its largest part is at most $n$, so it can be prepended by a new first part $2n - i > 0$. The new partition we obtain is a partition of $2n$. This sets up an injection from the set of partitions of all positive integers at most $n$ into the set of partitions of $2n$, and the proof follows $(**)$.
I am stuck at $(**)$. Since $\pi$ is a partition of $i$, and you add $2n-i$ to it, it will add up to $2n$, therefore $\pi$ will be a partition of $2n$. I don't understand why there is an injection (is it because we can do some manipulation and get a partition of $2n$ from the partition of $i$?) from the set of partitions of all positive integers at most n (why at most n?) into the set of partitions of $2n$. And how does it follow from there? How does this imply the above inequality?
I would appreciate any help or hints.
