If `0 lt alpha lt (pi)/(3)`, then prove that `alpha (sec alpha) lt (2pi)/(3).`

Explanation

To prove that if \( 0 < \alpha < \frac{\pi}{3} \), then \( \alpha \sec \alpha < \frac{2\pi}{3} \), we can follow these steps:

### Step 1: Understanding the Function
We start with the inequality we want to prove:
\[ \alpha \sec \alpha < \frac{2\pi}{3} \]
Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \). Thus, we can rewrite the inequality as:
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