Radial gradient corrupted during conversion from EPS10 into bigger JPEG

Question

I use Illustrator CS4 and Photoshop CS4 extended.
recently came scross rasteirizing of radial gradient when convert from AI CS4 file to EPS10 file and then buyer converts it into big 4000x4000 pxls JPEG. Never experienced that before.
It is visible only after conversion EPS into big JPEG.
Linear gradients are all smooth after conversion but radial is corrupted.
I clean my illustration before saving as EPS.
When I save my illustration AICS4 as EPS10,  I check:
-preview-transparent;
-transparency preset : high resolution 100%,
     -convert all strokes in outline,
     -convert text in outline (if I have it in my illustration);
-include CMYK PostScript in RGB file,
-Compatible Gradient and Gradient Mash Printing.
I check then EPS file for rasteirizing : window-links.
No sign of any rasteirization. All nice and smooth.
here, how looks corrupted radial gradient after conversion to big JPEG from small EPS10. Note, that on the bottom of image  linear gradient is almost ok.
What I am doing wrong?
When I upsize my AI CS4 file to 4000x4000pxls and then convert to EPS10, and then into JPEG of the same size, - Radial gradient is smoother, almost perfect. But the size of of EPS file is too big and out of acceptable limits.
Jim

Answer 1

Thank you for reply.
Now I see that I did not explain it properly.
Sorry for that.
I need to provide my illustration saved as EPS10 with small Jpeg for preview via internet to the client, the way he requires to have it. I do not know why he wants it that way.
After receiving my EPS file with small Jpeg, he is making from that EPS10 file a big JPEG for  his needs. It is what he wants.
When he is convertin' EPS10 file into big JEPEG, radial gradient becomes corrupted.
I followed his steps to try to undertstand what went wrong and got the same corrupted radial gradient in JPEG file, as you see on provided image.
What I want to know is how to accomplish what he wants without loosing quality.
Could you tell me, please, how to save illustration as EPS10 in Illustrator CS4 proper way.
Thank you.