Prove that variance of a portfolio cannot exceed variance of individual assets
Let $P = \alpha A + (1-\alpha) B$ where $A$ and $B$ are returns (random) from the two assets, and $P$ is their portfolio. 
Variance of portfolio $P$ can therefore be written as

\begin{eqnarray*} \sigma^2_P &amp; = &amp; \alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\text{Cov}(A, B) \\ &amp; \leq &amp;\alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\sigma_A\sigma_B \\ &amp; = &amp;(\alpha\sigma_A + (1-\alpha)\sigma_B)^2 \\ &amp; \leq &amp; (\alpha \max(\sigma_A, \sigma_B) + (1-\alpha)\max(\sigma_A, \sigma_B))^2 \\ &amp; = &amp; \max(\sigma_A^2, \sigma_B^2)\end{eqnarray*}