Proof of Limit Comparison Test

As 0 < c <∞ we can find out two positive and finite numbers, m and M, like m < c < M .

 Now, as  

we know that for large enough n the quotient a_n/b_n should be close to c and thus there must be a positive integer N like if n > N we as well have,

m < a_n / b_n < M

Multiplying by b_n provides

Mb_n < a_n < Mb_n

provided n > N .

Here now, if ∑b_n diverges then thus does ∑mb_n and so as mb_n < a_n for all adequately large n by the Comparison Test ∑a_n as well diverges. 

Similarly, if  ∑b_n converges then so does ∑Mb_n and as a_n < Mb_n for all sufficiently large n by the Comparison Test ∑a_n as converges.