 prove the inequality:

For all n∈N∗n∈N∗ and for all (x,y)∈(R+)2(x,y)∈(R+)2,

x+yn≤xn+ynnx+y​≤nx​+ny​




To prove the inequality:

\[
\frac{x + y}{n} \leq \frac{x^n}{n} + \frac{y^n}{n}
\]

for all \( n \in \mathbb{N}^* \) (positive natural numbers) and \( (x, y) \in (\mathbb{R}^+)^2 \) (positive real numbers), we will use **Jensen's inequality** applied to convex functions.

However, your expression seems incomplete or unclear. It looks like you intend to use convexity properties or inequalities related to powers, perhaps comparing the arithmetic mean and a mean related to powers.

Let's rewrite the inequality to something more standard or precise:

### Step 1: Standard Formulation

The inequality you might want to prove could be related to **power

Learn how to prove the inequality x + y^n / n ≤ x^n / n + y^n / n for all n in positive natural numbers and (x, y) in positive real numbers using Jensen's inequality and properties of convex functions.

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