False Proof
A student (A.K.) passed this “gem” on to me. How would you respond?
We know that x^{2} = (x + x + x + .... + x + x) i.e. x added to itself xtimes.
Now, take the derivative on both sides, to get:
2x = (1 + 1 + 1 + …. + 1 + 1) i.e. 1 added to itself xtimes.
Thus, 2x = x.
And, canceling the common term, we get 2 = 1.
This is a fact we always knew was true but could not prove until now!
