Proof: c * divergent sequence diverges

[tarheelborn]

Homework Statement

Suppose that {a_n} is a divergent sequence of real numbers and c \in R, c <> 0. Prove that {c*a_n} diverges.

Homework Equations

The Attempt at a Solution

I have attempted to solve the problem as a proof by contradiction, but am afraid I am leaving something out. Please confirm my proof is complete or prompt me to add. Thanks!

Proof is by contradiction. Suppose {c*a_n} is convergent. This means that |c*a_n - L| < e, for e > 0. Then there is N \in N such that |a_n - L/c| < e/|c|, n >= N. But this is the definition of limit of a sequence, so a_n converges. But this contradicts our problem statement so, in fact, a_n diverges. End of proof.

 

[LCKurtz]

You have the right idea, but it isn't written well. You need to use your observations to write your argument in reverse. Something like this:

Given ca_n $\rightarrow$ L we will show that a_n $\rightarrow$ L/|c|, which is a contradiction.

Suppose$\ \epsilon > 0$. Then there is N > 0 such that:

| ca_n - L| < |c|$\epsilon$

for all n > N. Therefore

|ca_n - L| = |c(a_n - L/c| = |c||(a_n - L/c| < |c|$\epsilon$

which gives, upon dividing that last inequality by |c|, | a_n - L/c| < $\epsilon$
for all n > N.