Usage of "rather than to"

I don't know if "rather than to" is correct or if there is a better turn of phrase:

Note that our aim is to show [...] rather than to run [...]

I can't begin to guess what it is about the basic structure X rather than Y that bothers OP, so all I can say there is it's perfectly normal English, in both spoken and written contexts.

The only other thing I'd add is that stylistically most speakers/writers would elide the second to...

to show rather than tell (8220 hits in Google Books)
to show rather than to tell (1060 hits)

As FumbleFingers says, expressions of the form "our aim is to show...rather than to run..." are common in English.

There are, however, a number of alternatives to "X rather than Y" that are likewise common in everyday English, including these:

X instead of Y [["I want mashed potatoes instead of french fries."]]

X, not Y [["I come to bury Caesar, not to praise him."]]

more for X than for Y [["Our website is geared more toward Conan the Barbarian obsessives than toward garden-variety fans of sword-and-sorcery fantasy."]]

The formulations "X rather than Y," "X instead of Y," and "X, not Y" have in common the idea of identifying or advancing one thing in place of another. To my ear, "X in place of Y" strongly implies a swapping out of choice Y in favor of choice X. "X rather than Y" expresses much the same type of preference as "X instead of Y," but it seems to me a little less clearly focused on a one-for-one replacement. For its part, "X, not Y" is the most abrupt and the most emphatic in asserting that choice X entails the exclusion of choice Y.

The "more for X than for Y" differs from the other three in promoting X in preference to Y without doing so to the exclusion of Y.

So depending on the kind of relationship you mean to establish between "showing" and "running" as possible aims in the original example, you may find that a preferential description works better than a "this in place of that" description.