Question

Q. show that if X and Y are sequences such that X converges to x≠ 0 and XY converges, then Y converges.

Answer 1

Answer on Question #64524 – Math – Real Analysis

Question

Show that if $X$ and $Y$ are sequences such that $X$ converges to $x \neq 0$ and $XY$ converges, then $Y$ converges.

Solution

Since $X$ converges to $x \neq 0$ there exists $K$ such that for all $n > K$: $x_n \neq 0$.

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 $X \neq 0$, by properties of limits, $Y$ converges to $\frac{z}{x}$.

Hence $Y$ converges.

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