Show that the set S = { x/|x| <=2} is a convex set 

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.

Learn how to prove the convexity of set S = { x | x/|x| <= 2 }. This problem involves verifying that any line segment between two points in the set remains within the set itself. Suitable for advanced undergraduate students and researchers interested in convex analysis.

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