Question

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

Answer 1

Let:

`a_n = 5+1/n`

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.