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Contemplate the fact that from every strictly monotonic
function f:R->R we can construct a metric on R which is similar
to the Euclidean metric.

d(x,y) = |f(x)-f(y)|

It's a metric because f is injective.

It's similar to the Euclidean metric because f is continuous and so is its inverse.

It's a metric because f is injective.

It's similar to the Euclidean metric because f is continuous and so is its inverse.

0 <= a <= x

0 <= b <= x

Contemplate the fact that |b-a| <= x.

0 <= b <= x

Contemplate the fact that |b-a| <= x.

_____

Prove that for a,b >= 0, sqrt(a+b) <= sqrt(a) + sqrt(b).

Square it.

Investigate the uniform convergence of f(n,x) = x/(1+nx), on
[0,1].

YES.

hint: 1 + nx >= nx

hint: 1 + nx >= nx

Investigate the uniform convergence of

f[n](x) = sqrt(n) * x * (1-x*x)^n on [0,1].

f[n](x) = sqrt(n) * x * (1-x*x)^n on [0,1].

No uniform convergence on [0,1].

Use: x[n] = 1 / sqrt(n)

If 0<M<1, then uniform convergence on [M,1].

Use: x[n] = 1 / sqrt(n)

If 0<M<1, then uniform convergence on [M,1].

Investigate the uniform convergence of

the series x / ( (1+nx)*(1+(n+1)x) ).

(x>=0)

the series x / ( (1+nx)*(1+(n+1)x) ).

(x>=0)

It does not converge uniformly on [0,oo).

It converges uniformly on [M,oo) for all M>0.

hint: f[n](x) = 1 / (1+nx); f[n](x)-f[n+1](x)

It converges uniformly on [M,oo) for all M>0.

hint: f[n](x) = 1 / (1+nx); f[n](x)-f[n+1](x)

f(n,x) = x^n / (n + x^n), x>=0.

Investigate the uniform convergence of f.

Investigate the uniform convergence of f.

It does not converge uniformly on [0,oo).

It does not converge uniformly on (1,oo).

It converges uniformly on [0,1].

It converges uniformly on [M,oo) for all M>1.

It does not converge uniformly on (1,oo).

It converges uniformly on [0,1].

It converges uniformly on [M,oo) for all M>1.

Given two complex numbers A and B, B != 0, write the equation
of a line passing through points A and A+B.

Im( (z-A)/B ) = 0

see page 188 in the palace notebook

see page 188 in the palace notebook

What are the three equivalent definitions of a line on the
complex plane?

1) Im( (z-a)/b ) = 0, where a,b are complex numbers, b != 0

2) A*Re(z) + B*Im(z) + C = 0, where A,B,C are real numbers, A and B are not both zero

3) |z-A| = |z-B|, where A,B are distinct complex numbers

2) A*Re(z) + B*Im(z) + C = 0, where A,B,C are real numbers, A and B are not both zero

3) |z-A| = |z-B|, where A,B are distinct complex numbers

Prove that for every complex number A,

it is false that lim cos(n*A) = 0.

it is false that lim cos(n*A) = 0.

Use: lim cos(2*n*A) = 0.

page 189 in the palace notebook

page 189 in the palace notebook