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PROOF 175

Prove.

Moving against the gradient decreases the value

On differentiable functions.

What you may use

A1|F − L| < ε (0<|t|<δ)
ε-δ definition of the limit (Proof 39)
A2a limit is at most one value
uniqueness of limits (Proof 130)
A3a(b + c) = ab + ac
distributive law
A4(ab)c = a(bc)
associativity of multiplication
A5ab = ba
commutativity of multiplication
A6a × 1 = a
multiplicative identity
A7a ≠ 0 ⟹ a·a⁻¹ = 1
multiplicative inverse
A8(a − b) + b = a
undoing subtraction
A9(−a)b = −(ab)
sign and products
A10a≠0 ⟺ |a|>0, a² = |a|²
absolute value and squares
A110<c ⟹ (a<b ⟺ ac<bc)
order axiom (positive multiplication)
A12a < b ⟹ a + c < b + c
order axiom (addition)
A13a ≤ b < c ⟹ a < c
transitivity of order — a < b includes a ≤ b
A14|ab|=|a||b|, |−a|=|a|
absolute value and products
A150 < a ⟹ |a| = a
meaning of absolute value
A16a ≤ |a|
absolute value as an upper bound
A170<a, 0<b ⟹ 0 < a·b⁻¹
quotient of positives
A180 < c ⟹ 0 < c/2 < c
half of a positive
A19a × 0 = 0, a − 0 = a
computation with 0
A20a + (−a) = 0
additive inverse
A21a + b = b + a
commutativity of addition
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

f is differentiable at x when the limit lim (t→0) [f(x+t) − f(x)]/t exists, and that value is written f′(x).

Proof.

Related labs

Gradient Descent lab →Derivative lab →