Hey, I was wondering if you can prove the following in the following way.

Given any infinite sequence, you can find a continuous function that interpolates each point in the sequence.

Proof (sketch):

Break the infite sequence into blocks of n, where n is finite. By a block of n I mean the first n terms, then the next n terms and so on. The union of these 'blocks' is the entire sequence.

Now, for each n block, we can applay the lagrangian interpolation formula to get a contious polynomial that intersects the n points. Then, you can do the same for the next block. And so on. Say we have 2 blocks, then we have 2 continuous functions. Now, we can turn them into one continuous function by keeping the parts of each function that intersect the points in its corospoding block and then erase the sections of the functions that are on the interval (n, n+1). Then define the new function in that area as the max of those 2 on the interval (n,n+1) if they don't intersect any more times in that interval. If they do, then you just apply the max on the intervals demarked by the intersections. In any area if the 2 functions overlap (not infinite intersection but both fill space above x axis), then just take the max.

Continue by induction.

Does this work. Please bare with me as I don't know a great deal about interpolation. If there is anonther theorem that states that you can interpolate a finite sequence with a continuous function, I would really appreciate learning about it.