Using the definition of convergence, how do you prove that the sequence {5+(1/n)}{5+(1n)} converges from n=1 to infinity?

1 Answer
Jun 29, 2018

Let:

a_n = 5+1/nan=5+1n

then for any m,n in NN with n > m:

abs (a_m-a_n) = abs( (5+1/m) -(5+1/n))

abs (a_m-a_n) = abs( 5+1/m -5-1/n)

abs (a_m-a_n) = abs( 1/m -1/n)

as n > m => 1/n < 1/m:

abs (a_m-a_n) = 1/m -1/n

and as 1/n > 0:

abs (a_m-a_n) < 1/m.

Given any real number epsilon > 0, choose then an integer N>1/epsilon.

For any integers m,n > N we have:

abs (a_m-a_n) < 1/N

abs (a_m-a_n) < epsilon

which proves Cauchy's condition for the convergence of a sequence.