Metamath Proof Explorer


Theorem xrge0tsms2

Description: Any finite or infinite sum in the nonnegative extended reals is convergent. This is a rather unique property of the set [ 0 , +oo ] ; a similar theorem is not true for RR* or RR or [ 0 , +oo ) . It is true for NN0 u. { +oo } , however, or more generally any additive submonoid of [ 0 , +oo ) with +oo adjoined. (Contributed by Mario Carneiro, 13-Sep-2015)

Ref Expression
Hypothesis xrge0tsms2.g G = 𝑠 * 𝑠 0 +∞
Assertion xrge0tsms2 A V F : A 0 +∞ G tsums F 1 𝑜

Proof

Step Hyp Ref Expression
1 xrge0tsms2.g G = 𝑠 * 𝑠 0 +∞
2 simpl A V F : A 0 +∞ A V
3 simpr A V F : A 0 +∞ F : A 0 +∞
4 eqid sup ran x 𝒫 A Fin G F x * < = sup ran x 𝒫 A Fin G F x * <
5 1 2 3 4 xrge0tsms A V F : A 0 +∞ G tsums F = sup ran x 𝒫 A Fin G F x * <
6 xrltso < Or *
7 6 supex sup ran x 𝒫 A Fin G F x * < V
8 7 ensn1 sup ran x 𝒫 A Fin G F x * < 1 𝑜
9 5 8 eqbrtrdi A V F : A 0 +∞ G tsums F 1 𝑜