What should be the value of $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$? $\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}$
Obviously, by using common-sense the answer is $0$. But I had thought of a different mathematical approach.
$Let:$
$x = \sqrt{0+\color{red}{\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0+\sqrt{0 ...}}}}}}}$
Since the $\color{red}{red}$ part also symbolies/equals to '$x$', therefore we can rewrite this as:
$x = \sqrt{0 + x} \\ \text{Squaring, both the sides} \\ x^2 = 0 + x \\ x^2 - x = 0 \\ x(x-1) = 0 \\ \text{Hence, the solutions are 0 and 1}$
$1$ seems to be impossible, but it clearly satisfies my assuming 'x'. So is my assumption wrong ? or should is just ignore this fact.
 A: There are two useful responses that come to mind, both of which appeared in comments as I was typing this:

*

*Firstly, not all implications go both ways. As @TheSilverDoe said in a comment, just because a number has some property does not mean it is the only number with that property. Squaring an equation is an example which can introduce alternative solutions as that comment shows: $x = 0 \implies x^2 = x$ which has solution $x=1$ too. Another classic error is multiplying by $0$;
$x = 5 \implies 0 \times x = 0$
which is solved by $x = 7$ as well as $x=5$.

*Secondly, as @JMorawitz emphasises, you have to make sure that mathematical expressions (with unique values you can work out) are well-defined objects, rather than being ways of defining equations which may have multiple solutions. (There's a hazy grey-area of multivalued functions, too, but let's not worry about handling these.) With infinite expressions, "things with dots in", this is particularly likely to be an issue.
The usual mathematical formulation of infinite nested expressions (that avoids any confusion) is given by defining the result as the limit of a sequence. The above comment highlights one standard way to give a clear definition of what your expression means, namely we take a sequence
$$a_{n+1} = 0 + \sqrt{a_n}$$
along with some initial value $a_0$, and ask what the limit of $a_n$ as $n\to\infty$ is. This depends on the initial value $a_0$. In fact,
$$
\lim_{n\to\infty} a_n = \begin{cases}0 & \text{if }a_0 = 0 \\ 1 & \text{if }a_0 \neq 0\end{cases}
$$
So you need to clearly define what you mean, perhaps by adopting a convention like $a_0 = 0$ since that's what you get if you 'stop writing roots' when 'filling in the dots'.

