Given the string an=(n+1)/(5n+2), verify that the limit is 1/5 using epsilon-delta theory.

We'll have to show that for any positive number epsilon,
there is N = N(epsilon), so that:

|an - 1/5| <
epsilon for any n > N(epsilon)

We'll substitute an
by it's given expression:

|(n+1)/(5n+2) - 1/5| =
|(n+1-5n-2)/5(5n+2)| = 3/5(5n+2)

We'll get the
inequality:

3/5(5n+2) <
epsilon

3 < 5 epsilon (5n +
2)

We'll remove the
bracktes:

3 < 25*epsilon*n + 10
epsilon

We'll subtract 10
epsilon:

25*epsilon*n > 3 - 10
epsilon

n > (3 - 10 epsilon)/25
epsilon

N(epsilon) = 3/25epsilon -
2/5

For all terms that have n >
N(epsilon) = 3/25epsilon - 2/5,

|an - 1/5| <
epsilon