Question

If (a_n) converges to a and (b_n) converges to b show that (a_n+or - b_n) converges to (a+or - b)

Answer 1

For given ε > 0, there exists a positive integer $N_1$ such that

$n> N_1$ implies |$a_n-a$ | < ε/2.

Moreover, there exists a positive integer $N_2$ such that

$n>N_2$ implies |$b_n-b$ | < ε/2.

Let N := max{$N_1,N_2$ }. If n > N, then by the triangle inequality we have

$|(a_n ± b_n) − (a ± b)| ≤ |a_n − a| + |b_n − b| < \dfrac{ε} {2} +\dfrac {ε}{ 2} = ε$

This completes the proof. If $(a_n)_{n=1,2,...}$ is a convergent sequence, then the above theorem tells us that

$lim_{n→∞} (a_{n+1} − a_n) = lim_{n→∞} a_{n+1} − lim_{n→∞} a_n = 0.$