While it may be too early to say that inflation is proved, I claim that the case for inflation is compelling. It is hard to even conceive of an alternative theory that could explain the basic features of the observed Universe. Not only does inflation produce just the kind of special bang that matches the observed Universe, but quantum fluctuations during inflation could have produced nonuniformities which served as the seeds of cosmic structure. These nonuniformities can be observed directly in the cosmic background radiation, with an amplitude of about one part in 100,000. So far the measurements of the spectrum have been beautifully consistent with the predictions of inflation, although it must be admitted that nonuniformities created by cosmic strings are also consistent with the observations. Cosmic strings, however, cannot explain the large-scale homogeneity or the flatness of the Universe.
While the case for inflation is strong, it should be stressed that inflation is really a paradigm and not a theory. The statement that the Universe arose from inflation, if it is true, is not the end of the study of cosmic origins - it is in fact closer to the beginning. The details of inflation depend upon the details of the underlying particle physics, so cosmology and particle physics become intimately linked together. While I cannot see any viable alternative to the general idea of inflation, there is still much work to be done before a detailed picture is established. And I suspect that there is room for many new important ideas.
The expanding sphere illustrates the solution to the flatness problem in inflationary cosmology. As the sphere becomes larger, its surface becomes flatter and flatter. Similarly the inflation of space causes it to become geometrically flat, and general relativity implies that the mass density of a flat universe must equal the critical value.