formal proof

do your own homework brainlet

>>8771757
proof by contradiction. Suppose 5 is not real. Then sqrt(5) could not be real either. But sqrt(5) is real. Therefore 5 is real.

>>8771774
Lol what. How does sqrt have to be real?

>>8771774
This is so stupid, you're using so much more information past the thing op is trying to prove.

You're using the corollary "the square root of a real positive number is real" which requires a loot of machinery to prove; for example the square root of a real negative number is not real

In a larger sense, any answer to OP's question is retarded, since he is asking for a proof of a definition.

We define the set of real numbers to include 5, therefore 5 is a real number.

>>8771954
>We define the set of real numbers to include 5, therefore 5 is a real number.
this is false btw

>>8771757
The following sequence of rationals converges to 5:
the sequence is 5, 5, 5, 5, 5....
therefore 5 is real

>>8771774
suppose -5 is not real
then sqrt(-5) could not be real either, which is true
hence, -5 is not real

>>8772108
/thread

>>8771757
You can build R as the quotient space of all classes of Cauchy sequences in Q with the relationship of having the same limit
You can build Q as the quotient space of all classes of ordered couples of whole numbers (a,b) with (a,b)~(c,d) if ad=bc
You can build Z as N + all the additive inverses of N
N is defined as a set with these properties:
0 is a natural number.
Every natural number has a successor.
0 is not the successor of any natural number.
If the successor of x equals the successor of y, then x equals y.
If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number

S(0)=1, S(1)=2, S(2)=3, S(3)=4, S(4)=5
5 is a natural number => 5 is a whole number => 5 is a rational number => 5 is a real number

>>8772110
brainlet.

>>8772480
tl;dr

>>8772480
>0 is a natural number.
>Every natural number has a successor.
>0 is not the successor of any natural number.

u fukn w0t

>>8772544
nevermind, i am retard

i thought the second clause of the definition was "every natural number is a successor (of a natural number)"

pls keep me out of academia

>>8772110
>one way implication

>>8771757
consider the sequence 5,5,5,5,5.....
QED