Proposition 4

Tag 085Y, permanent. Cite it as [Bourbaki, Tag 085Y].

Functions of a Real Variable, chapter I, § 4. CONVEX FUNCTIONS OF A REAL VARIABLE

Let $H$ be a set of convex functions on an interval $I \subset \mathbf{R}$ ; if $\mathfrak{F}$ is a filter on $H$ which converges pointwise on $I$ to a finite real function $f_0$ , then this function is convex on $I$ .

To see this it suffices to pass to the limit along $\mathfrak{F}$ in the inequality (1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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