# Proof with natural numbers and sequences of functions

## 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...

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LCKurtz
Homework Helper
Gold Member

## 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 fi near it and using the fact that they are Cauchy.

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

LCKurtz
Homework Helper
Gold Member
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 fi(n). For each i, fi is a function on the integers, that is, a sequence. It doesn't make sense to talk about fi converging pointwise for a given i. For each i, {fi(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 fi → g pointwise and you might ask yourself whether fi → g uniformly on N, given the statement of the problem.

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?