Convergence of {c_n} to c when {c_n - c} converges to zero

[Repetit]

Homework Statement

Prove that the sequence {c_n} converges to c if and only if the sequence {c_n - c} converges to zero.

Homework Equations

The Attempt at a Solution

First I prove that the convergence of {c_n - c} to zero implies that {c_n} converges to c by (all limits take n to infinity):

The convergence of {c_n - c} means that

lim[c_n - c] = 0

Using the sum property of limits we get

lim[c_n] - lim[c] = lim[c_n] - c = 0

and therefore

lim[c_n] = c

I am in doubt about how to prove the "only if" part. I am thinking of assuming that if {c_n - c} converges to another number a which is no zero, and then proving that under these circumstances {c_n} cannot converge to c, but I am in doubt if this is the correct way to do it.

Thanks in advance.
René Petersen

 

[JonF]

If you write out the ε definition of convergence, this is trivial. Remember you can always add or subtract zero, and group terms however you like.