<s>[INST] Induction $(1+\frac{1}{x^n})(1+\frac{1}{y^n}) \geq (1+2^n)^2$ How to prove this inequality using Induction (or any simpler method):
Let (x,y) be real positive numbers, so that x+y=1; and n an integer:
Prove this:
$\begin{align}(1+\frac{1}{x^n})(1+\frac{1}{y^n}) \geq (1+2^n)^2\end{align}$
 [/INST] Replace $ y$ with $1-x$.
Hint: Show that $ \frac{1}{4} \geq x(1-x)$.
Hint: Show that $\frac{1}{x^n} \times \frac{1}{(1-x)^n} \geq 2^{2n} $.
Hint: Show that $ \frac{ 1}{x^n} + \frac{1}{(1-x)^n} \geq 2 \times 2^n$ using the technique AM-GM.
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