fake proofs - Can this really happen?

Yesterday, while scribbling on the back page of my maths copy, I accidentally came across this

If $ x\in R\ $ and $ x+x+x+.....\infty = m $ where $ m $ is any real number.

Then we can write

$ x+x+x+.....\infty = m $

$\Rightarrow $ $ x+m=m $

$ \Rightarrow $ $ x=0 $

In the same way we can prove that $ 6=0 $ or $ 7=0 $

But how is this really possible? Please explain. I am in 10th standard.