Prove that sqrt(5) is irrational?

My work

Suppose sqrt(5) is rational

5 = (m/n)^2

5 = m^2/n^2 or m^2 = 5n^2

**Since m^2 is a multiple of 5, m must also be a multiple of 5? - proof?**

m = 5k for some int k

5n^2 = (5k)^2

5n^2 = 25k^2

n = 5k^2

n and m are both multiples of 5 this contradicts the assumption sqrt is irrational

end of work

My professor didnt like what i wrote in bold. I suppose she wanted me to write a proof out for this? how would I proceed? Is this broken up into cases?

thank you

ps. Im sorry I have so many topics in this area. Im trying my hardest to get by :(