Prove that variance of a portfolio cannot exceed variance of individual assets When reading on Markowitz's portfolio theory, I stumbled across the fact that in a market with two risky assets, if no short selling is not allowed, the variance of a portfolio consisting of the two assets cannot exceed the variances of the risky assets individually. That is:

$${\sigma _p}^2 \le \max \{ {\sigma _A}^2,{\sigma _B}^2\} $$
Where A and B are two different assets.

Could you kindly prove this statement, and possible provide some intuition for why this is the case?
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*}