Examples of algorithms and proofs that seem correct, but aren't

For algorithms that seem correct, but aren’t, look at any security vulnerability in decently audited software. A nice example is https://crocs.fi.muni.cz/public/papers/rsa_ccs17

Also, see ”Nearly All Binary Searches and Mergesorts are Broken” (https://research.googleblog.com/2006/06/extra-extra-read-all...), which is caused by the difference between mathematical integers and computer ‘ints’.