Proof with natural numbers and sequences of functions

[davitykale]

Homework Statement

For every epsilon > 0, there exists an N\in N such that, for every j >= N, |f(i,n) - g(n)|<epsilon for every n\in N. In addition, for every fixed j\in N, (f(i,n)) converges. Prove that (g(n)) converges.

Homework Equations

f: N x N --> R, g: N --> R

The Attempt at a Solution

I'm not sure what to do, especially since we're dealing with domains of N and not R...

 

[LCKurtz]

Homework Statement

For every epsilon > 0, there exists an N\in N such that, for every j >= N, |f(i,n) - g(n)|<epsilon for every n\in N. In addition, for every fixed j\in N, (f(i,n)) converges. Prove that (g(n)) converges.

Homework Equations

f: N x N --> R, g: N --> R

The Attempt at a Solution

I'm not sure what to do, especially since we're dealing with domains of N and not R...

Do you know about the Cauchy criteria for a sequence to be convergent? Prove g is a cauchy sequence by getting the f_i near it and using the fact that they are Cauchy.

 

[davitykale]

Does the "fixed i\in N" imply that f(i,n) converges pointwise? I only know the cauchy criterion for uniform convergence

 

[LCKurtz]

Does the "fixed i\in N" imply that f(i,n) converges pointwise? I only know the cauchy criterion for uniform convergence

I prefer the notation f_i(n). For each i, f_i is a function on the integers, that is, a sequence. It doesn't make sense to talk about f_i converging pointwise for a given i. For each i, {f_i(n)} is a convergent sequence. For example, f₁ might be the sequence:


f₁(1) = 1
f₁(2) = 1/2
f₁(3) = 1/4
f₁(4) = 1/8
...
f₁(2) = 1/2ⁿ

which is a convergent sequence.

What does make sense is to say f_i → g pointwise and you might ask yourself whether f_i → g uniformly on N, given the statement of the problem.

 

[davitykale]

Is this something like what you meant:

|f(i,n)−f(i,m)| = |f (i, n) − g(n) + g(n) − g(m) + g(m) − f (i, m)|
≤ |f (i, n) − g(n)| + |g(n) − g(m)| + |g(m) − f (i, m)|

Or does what I'm doing make any sense at all in terms of solving the problem?