Is this a valid proof?

Problem: Prove; If lim an < lim bn, then there exists N that belongs to the natural numbers such that n >= N implies an < bn

My solution:

Let lim an = L and lim bn = M. Assume L < M. Then for everyεthere exists N_{1}such that n >= N1 implies L -ε< an < L +ε.For everyεthere exists N2 such that n >= N2 implies M -ε< bn < M +ε. Let N = max{N_{1}, N_{2}}

=> L -ε< an < L + εM -

ε < bn <M+ ε

Assume an >= bn

Then L -ε < M -ε < bn <= an.

an < L +ε => L -ε < bn < L +ε => |bn - L| <ε => L=M which is a contradiction, thus an must be less than bn.