calculus - Prove or disprove: Two statements about Cauchy-sequences

Prove or disprove:

1. Every cauchy-sequence in $\mathbb{R}$ includes a subsequence which is monotonic.


2. Every monotonic increasing cauchy-sequence in $\mathbb{R}$ converges to its supremum.

1. I would say it's true because a main attribute of cauchy-sequences is that its sequences always get smaller and smaller with each other, so each one will be monotone.




2. I say it's false but I cannot reason it :p

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What do you think?