Imagine we live in a universe where there are only two optimal portfolios available.

• $P_o$ with a $0\%$ mean return and
• $P_1$ with a $1\%$ mean return

Given that
\begin{align}\mathbf{w} = \mathbf{g} + \mathbf{h}\mu_p
\end{align}
where $g$ and $h$ are constant column vectors, we then have
\begin{align}
\mathbf{w} &= \mathbf{g} \tag{0%}\\
\mathbf{w} &= \mathbf{g} + \mathbf{h} \tag{1}
\end{align}
We can buy a mix of $P_o$ and $P_1$ to create a new portfolio $P$, but must remain feasible. Therefore consider investing in $P_o$ and $P_1$ as follows

• Invest $\mu_p$ in $P_1$
• Invest $1\; – \mu_p$ in $P_o$

The new portfolio is feasible and has weights
\begin{align}
\mathbf{w_p} &= \mu_p(\mathbf{g}+\mathbf{h})+ (1 \; – \mu_p)\mathbf{g}\\
& = \mathbf{g} + \mathbf{h}\mu_p
\end{align}
as required. $P_o$ and $P_1$ generate the whole family of frontier portfolios. Any two frontier portfolios would work to generate the whole family of frontier portfolios.

Two Fund Theorem

The whole family of frontier portfolios can be generated by any two distinct frontier portfolios $P_A$ and $P_B$.