Show that every prime power if deficient.

Question

If $n$ is a positive integer, we say that n is deficient if $\sigma(n)<2n$.

The answer is given by:

Can someone please explain the two yellow marks? And also the connexion between the first and second mark?

Answer 1

First one: Start with $2\leq p$. Multiply both sides with $p^k$ to get $2p^k\leq p^{k+1}$. Finally, note that subtracting $1$ from a number makes it strictly smaller, so we get $$ 2p^k-1<2p^k\leq p^{k+1} $$ For the second one, take what we had in the first one, and add $p^{k+1} - 2p^k$ to both sides: $$ 2p^k - 1 + p^{k+1} - 2p^k < p^{k+1} + p^{k+1} - 2p^k $$ Clean up the expressions on both sides, and you get exactly $p^{k+1} - 1<2(p^{k+1} - p^k)$.