We consider an extension of the infinitary lambda calculus by Kennaway et al., with zero, successor, and conditional, and a type system akin to Goedel's system T. For terms that can be typed in this system, we define the Global Trace Condition (GTC), adapting the concept from Brotherston and Simpson's Cyclic Proofs, and show that any infinite reduction of a well-typed term satisfying the GTC is strongly convergent. As a corollary, we obtain the proof that any closed term of type Nat reduces to some numeral through any reduction by levels. We argue that the Church-Rosser in the limit holds for our calculus and, when restricted to regular terms, the calculus defines exactly the total functions defined in Das's Cyclic System T (an infinitary version of System T without $λ$), and hence in Goedel's System T.

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