A sequence with a limit that is a real number is called a convergent sequence.

A sequence which does not converge is said to be divergent.

Find two divergent sequences like \{x_k\},\{y_k\}\{x_k\},\{y_k\} such that \lim_{k \rightarrow \infty}|y_k-x_k|=0\lim_{k \rightarrow \infty}|y_k-x_k|=0 .

Notice that we know \forall k \in \mathbb N \space 0 \lt |y_k-x_k|\forall k \in \mathbb N \space 0 \lt |y_k-x_k|

Note ( For those who ask about my try ) : There is nothing to try! If my try was successful, I wouldn’t be asking this question.

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2 Answers

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x_n=(-1)^n+\frac1n\quad\text{and}\quad y_n=(-1)^n.

x_n=(-1)^n+\frac1n\quad\text{and}\quad y_n=(-1)^n.

It’s not clear to me why there is “nothing to try”; you can certainly try to think of two divergent sequences that have a minor (or no) difference.

For example, take x_k=y_k=kx_k=y_k=k, or less trivially, x_k=kx_k=k and y_k=k+\frac1ky_k=k+\frac1k