Error in my proof? What is wrong in this proof. It seems correct to me but still doesn't make proper sense.
$$\sqrt{\cdots\sqrt{\sqrt{\sqrt{5}}}}=5^{1/\infty}=5^0=1$$
EDIT
So does this mean that
$5^{1/\infty} = 1$
$(5^{1/\infty})^\infty = 1^\infty$
But according to me, $1^\infty$ is indeterminate.

Using $1/\infty$ doesn't make sense, as already stated, because ∞ is not a number, it is a concept. I believe this is the closest thing to what you're really asking.
$$\lim_{n\to\infty}5^{1/n}=1$$
This is true. n never actually reaches infinity, but we find what happens as it gets closer and closer. And it does, indeed, approach 1.  
This is because as n gets larger and larger, 1/n gets smaller and smaller. It is known as infinitesimal, essentially zero. It's a little more complicated than just zero (if you're interested, learn calculus, specifically limits and infinitesimals) but for the purposes of this, it's as good as zero. And because $5^{0}=1$, the limit shown above is equal to 1.
Edit: I'd also like to add that using infinity as a number is a bad idea.
$$1+\infty = \infty$$
$$2+\infty = \infty$$
$$1+\infty = 2+\infty$$
$$1 = 2$$
