I notice that most of the answers are cute and humorous, so I may incur the wrath of the forum by posting an answer that is neither cute nor humorous.
I would have to say that most informal proofs are not valid, but there are formal proofs that come to the same conclusion, so the conclusion is correct, but informal proofs tend to be incomplete and lack rigor.
Cantor's diagonal argument is a good example. How long will it take to get a seventh grader to understand something like this, http://us.metamath.org/mpegif/ruc.html ? I don't even know if that is right.
Yet, most seventh graders will understand vsauce's 2 minute explanation of the theory. But it is incomplete. https://www.youtube.com/watch?v=s86-Z-CbaHA&feature=youtu.be&t=4m35s
Vsauce is correct when he says (paraphrased) "If there is a 1 to 1 correspondence between the two, then we can match one whole number to each real number, then the sets are the same size".
The hidden assumption is that we must use the whole numbers in order of magnitude. Which is not true.
So he is wrong when he says "We have used up every single whole number, the entire infinity of them, and yet we can still come up with more real numbers." The truth is that he did not need to "use up every single whole number".
To avoid this, all he had to do is match a subset of the whole numbers with his purported list of all real numbers. Like this
$4 \to r_1$
$44 \to r_2$
$444 \to r_3$
$4444 \to r_4$
and so on.
Then vsauce can generate another infinite list of real numbers that were not on his first list and again we can match them to a subset of the whole numbers like this
$14 \to rr_1$
$144 \to rr_2$
$1444 \to rr_3$
$14444 \to rr_4$
and so on.
And we can repeat this a countable infinite number of times, while vsauce can repeat his operation an uncountable infinite number of times.
But that is what vsauce needs to prove, and he can only do it with the formal proof, and that will take a very long and involved video.