To show that the set $S = \left\{ x \mid \frac{x}{|x|} \leq 2 \right\}$ is a convex set, we need to verify that for any two points $x_1$ and $x_2$ in $S$, the line segment connecting these points is also contained within $S$. Mathematically, this means we need to show:

$\text{If } x_1, x_2 \in S \text{ and } \lambda \in [0, 1], \text{ then } \lambda x_1 + (1 - \lambda) x_2 \in S.$

Step 1: Express the Condition for $x_1$ and $x_2$ in $S$

For $x_1 \in S$ and $x_2 \in S$, we have: $\frac{x_1}{|x_1|} \leq 2 \quad \text{and} \quad \frac{x_2}{|x_2|} \leq 2.$

Step 2: Consider a Point on the Line Segment Between $x_1$ and $x_2$

Let $y = \lambda x_1 + (1 - \lambda) x_2$ for some $\lambda \in [0, 1]$. We want to check if $y$ belongs to $S$, which requires that: $\frac{y}{|y|} \leq 2.$

Step 3: Calculate $|y|$

The magnitude $|y|$ is given by: $|y| = \left| \lambda x_1 + (1 - \lambda) x_2 \right|.$

By the triangle inequality, we have: $|y| \leq \lambda |x_1| + (1 - \lambda) |x_2|.$

Step 4: Check the Condition for $y$ to Belong to $S$

We need to show: $\frac{\lambda x_1 + (1 - \lambda) x_2}{|y|} \leq 2.$ Since: $\frac{x_1}{|x_1|} \leq 2 \quad \text{and} \quad \frac{x_2}{|x_2|} \leq 2,$ it follows that: $\lambda \frac{x_1}{|x_1|} + (1 - \lambda) \frac{x_2}{|x_2|} \leq 2.$

Thus, by the convexity of the scalar multiplication and the linearity of the operations involved, the weighted average $\frac{\lambda x_1 + (1 - \lambda) x_2}{|y|}$ is less than or equal to 2. Therefore, $y \in S$.

Conclusion

Since any convex combination $y = \lambda x_1 + (1 - \lambda) x_2$ with $\lambda \in [0, 1]$ lies within $S$ whenever $x_1, x_2 \in S$, the set $S = \left\{ x \mid \frac{x}{|x|} \leq 2 \right\}$ is indeed a convex set.

---

Would you like more details on any part of this proof or have any questions? Here are some related questions that might interest you:

1. What are the properties of convex sets in general?
2. How does the triangle inequality apply in convexity proofs?
3. Can you explain the concept of a convex combination in more detail?
4. What are some examples of convex sets in different dimensions?
5. How does convexity relate to optimization problems?

Tip: When proving convexity, always remember that the key is to show that any line segment between two points in the set remains entirely within the set.