Answer to Question #122828 in Real Analysis for NILAM JYOTI DAS

Question

Suppose that {xn} is a convergent sequence and {yn} is such that for any ε>0, there exists M such that |xn−yn|< ϵ for all n≥M. Show that {yn} is a convergent sequence.

Answer 1

Given that for any $\epsilon>0$ , there exists M such that $|x_n-y_n|<\epsilon$ for all $n\geq M$ ,

$\implies -\epsilon < x_n-y_n < \epsilon \implies x_n-\epsilon<y_n<x_n+\epsilon$ for all $n\geq M$ .

Now, given $\{x_n\}$ is a convergent sequence, $\implies \exist M : |x_n|<\epsilon_1 \ \forall n\geq M_1$ .

So, $\exist M_2:|y_n|<max\{\epsilon,\epsilon_1\} \ \forall \ n\geq M_2 = max\{M,M_1\}$ .

Hence $\{y_n\}$ is a convergent sequence.