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.
 A: I guess you mean this:
$$\lim_{n\to\infty}5^{1/2^n}=1$$
A: Let $x=\sqrt{\sqrt{...\sqrt{5}}}$.
$$x^2 = \sqrt{\sqrt{...\sqrt{5}}} = x$$
$$x = 0 \,  or \, 1 $$
Verify:$$\sqrt{5}=2.24$$
$$\sqrt{\sqrt{5}}=1.50$$
$$...$$
From the pattern we can see $x$ tends to $1$, so $x = 0$ is rejected.
A: What seems incorrect is because the language seems informal. Formally, one would write 
$$
x_0=5, \quad x_n = \sqrt{x_{n-1}}
$$
to describe the iterations of the square roots. One computes easily that 
$$
x_n = x_0^{1/{2^n}}
$$
and since $x_0^{1/n} \to 1$ for all $x_0 > 0$, so does $x_n$. (You can prove this by showing that the logarithm of the sequence goes to $0$ ; this is trivial.)
Hope that helps,
A: $\sqrt{...\sqrt{\sqrt{\sqrt{5}}}}=\lim x_n$, where $x_n=5^{(1/2^n)} (n\in\mathbb{N})$
A: $$n=\sqrt{...\sqrt{\sqrt{\sqrt{5}}}}$$
Since there's an infinite cascade of roots, we can add 1 (or as many as we want):
$$n=\sqrt{...\sqrt{\sqrt{\sqrt{\sqrt{5}}}}}$$
Replacing by the $n$ from the first equation:
$$n=\sqrt{n}$$
or
$$n^2=n$$
Since square roots of a number greater than 1 never can become less than 1, we can discard the solution $n=0$, so we can divide both sides by n, and 
$$n=1$$
A: $$\lim_{n\to\infty}x^{1/2^n}=1$$ because the exponent approaches zero
A: Note that $\lim_{n\to \infty}x_n^{y_n}=(\lim_{n\to \infty}x_n)^{\lim_{n\to \infty}y_n}$
A: 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.
$$1+\infty = \infty$$
$$2+\infty = \infty$$
$$1+\infty = 2+\infty$$
$$1 = 2$$
