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It is stated in this section that the first condition for integrability is always satisfied: If [math]f[/math] is bounded on [math][a,b][/math], then there exists a real number [math]J[/math] such that [math]L_\sigma \leq J \leq U_\tau[/math] for any two partitions [math]\sigma[/math] and [math]\tau[/math] of [math][a,b][/math].

• Show that one such number is the least upper bound of all the lower sums [math]L_\sigma[/math]. (This number is called the lower integral of [math]f[/math] from [math]a[/math] to [math]b[/math].)
• Show that another possibility is the greatest lower bound of all the upper sums [math]U_\tau[/math]. (This number is the upper integral of [math]f[/math] from [math]a[/math] to [math]b[/math].)
• Show that [math]f[/math] is integrable over [math][a,b][/math] if and only if the lower integral from [math]a[/math] to [math]b[/math] equals the upper integral, and that if the lower integral equals the upper then their common value is [math]\int_a^b f[/math].