# Bound on unit vectors

could someone help me with this simple problem. As always with homework, hints are specially welcome.

Let $v=(v_1,v_2)$ be a two-dimensional unit vector with complex coefficients. If $|v_1|<a$ and $|v_2|<a$ then $|v_1|+|v_2|\geq \frac{1}{a}$.

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$$|v_1|+|v_2|\geq \frac{|v_1|}{a}|v_1|+\frac{|v_2|}{a}|v_2|\geq \frac{|v_1|^2+|v_2|^2}{a}=\frac{1}{a}$$