Question

Suppose that (𝑥∝)∝𝜖𝐽 ⟶ 𝑥 in X and (𝑦∝)∝𝜖𝐽 ⟶ 𝑦 in Y. Show that

(𝑥𝛼 × 𝑦𝛼) → 𝑥 × 𝑦 𝑖𝑛 𝑋 × 𝑌.

Answer 1

(𝑥𝛼 × 𝑦𝛼) → 𝑥 × 𝑦 𝑖𝑛 𝑋 × 𝑌

proof considering the limits

lim $\alpha\to1$ $(x^\alpha)^\alpha = X$ and lim $\alpha$$\to1$ $(y^\alpha)^\alpha = Y$

lim $\alpha$$\to1$ $(x^\alpha)^\alpha = \alpha x ,$ lim $\alpha$$\to1$ $(y^\alpha)^\alpha = \alpha Y$

since $\alpha\to1$ then (𝑥𝛼 × 𝑦𝛼) → 𝑥 × 𝑦 𝑖𝑛 𝑋 × 𝑌