show that 0x = 0 for all x ∈ R

Use a completeness property of the real number system to show that 2
has a positive square root.

Use the Induction Theorem to show that for any natural number m,
the sum m+n and the product mn are again natural for any natural number n.
Thus N is closed under addition and multiplication, and consequently so is Z.
(b) Which of the field axioms continue to hold for the natural numbers?
(c) Which of the field axioms continue to hold for the integers?

Use the Intermediate Value Theorem to show that 2 has a positive
square root.

Let f : [0, 1] −→ [0, 1] be continuous. Use the Intermediate Value
Theorem to show that f(x) = x for some x ∈ [0, 1].

Let a and b be real numbers with a < b. Suppose that f : [a, b] −→ R
is continuous and that f is differentiable on the open subinterval (a, b). Use
the Mean Value Theorem to show that if f
′ > 0 on (a, b) then f is strictly
increasing on [a, b].

Write down any three specific nonzero vectors u, v, w from R
3 and any
two specific nonzero scalars a, b from R. Compute u+v, aw, b(v+w), (a+b)u,
u + v + w, abw, and the additive inverse to u.

Simplify - 8/19+-3/57​

Divide segment  AB of length 7 cm into a ratio 3:2​

What is the percentage decrease in an interval containing root after iteration is applied by Bisection Method? 30% 40% 50% 20%​

The cost price of 25 articles is equal to the selling price of 20 articles. What is the gain or loss percent?​