Chinese Remainder Theorem not necessarily coprime moduli (part of proof)

The usual Chinese Remainder Theorem requires that the moduli be pairwise relatively prime. A modification states that a given system of linear congruences $x\equiv a_i\pmod{m_i}$ is solvable if and only if $a_i\equiv a_j\mod{(n_i,n_j)}$ for all $i\neq j$.

I've looked around and I've got it narrowed down to this:

The system consisting of the congruences $x\equiv a\pmod{p}$ and $x\equiv b\pmod{p^2}$ is solvable if and only if $a\equiv b\pmod{p}$.

What I can't seem to find is how $a\equiv b\pmod{p}\Rightarrow$ a solution exists.

If $a\equiv b\pmod{p}$, then $x=b$ is a solution :-)
More generally, if we're given a prime number $p$ and integers $a$ and $b$ with $a\equiv b\pmod{p}$, the solutions to the system \begin{align} x&\equiv a\pmod{p}\\ x&\equiv b\pmod{p^2} \end{align} are precisely $$x\in\{b+kp^2\mid k\in\mathbb{Z}\}.$$