Answer to Question #115534 in Real Analysis for Karima Nisa

Question

Show that if X and Y are sequences such that X and X þ Y are convergent, then Y is convergent.

Answer 1

Since X converges to 𝑥 ≠ 0 0 there exists 𝐾 such that for all 𝑛 > 𝐾: 𝑥_n ≠ 0

Let 𝑙𝑖𝑚 𝑋𝑌 = 𝑧.

𝑋 converges to 𝑥 for all 𝑛 greater than a certain number and 𝑋𝑌 converges to 𝑧

for all 𝑛 greater than some number.

Since 𝑌 = $\frac{XY}{X}$ 𝑖𝑓 𝑋 ≠ 0, by properties of limits, 𝑌 converges to $\frac zx$ .

Hence 𝑌 is convergent .